The influence of variable operating conditions upon the general multi-modal Weibull distribution

Reliability Engineering and System Safety 64 (1999) 383–389

The influence of variable operating conditions upon the general multi-modal Weibull distribution Marko Nagode*, Matija Fajdiga University of Ljubljana, Faculty of Mechanical Engineering Asˇkerceva 6, 1000 Ljubljana, Slovenia Received 16 October 1997; received in revised form 10 July 1998; accepted 17 September 1998

Abstract For the design spectrum prediction that should be realized within the expected service life, the influence of variable loading conditions is of paramount importance. Further, the results of the measurements must be properly extrapolated and the scatter of loading spectra has to be determined to assure reliable service life prediction. To model load ranges, a general multi-modal Weibull distribution function has recently been proposed. Until now it has been verified only for fixed operating conditions. The scope of this article is to prove that the same distribution model holds for the case of variable operating conditions, too. The influence of variable operating conditions upon the distribution function is demonstrated by a few examples attained by analysing loads acting upon a structure of a fork-lift. 䉷 1999 Elsevier Science Ltd. All rights reserved. Keywords: Weibull distribution; Variable operating conditions; Service life prediction

1. Introduction The service life of a structural component depends decisively on the loading conditions in service. For fatigue and reliability analyses, it is therefore essential to define the design spectrum that should be realized under all possible operating conditions within the expected service life. The main parameters that influence the design spectrum are structural behaviour, usage and operating conditions [1]. The structural behaviour depends on the design and dynamic properties and can not be changed during operation. The only parameters that can be varied are usage and operating conditions. At this point a distinction between fixed and variable operating conditions should be clarified. Fixed operation conditions are the ones that do not change during service life. On the contrary, variable operating conditions do change during operation. If the data about the usage and operating conditions are known, the design spectrum can be assembled from loading spectra obtained by field measurements at all possible operating conditions. A general multi-modal probability density function (p.d.f.) has been suggested [2] to model loading spectra. It consists of m Weibull distributions. The proposed

* Corresponding author. Tel.: ⫹ 386-(0)61-1771-503, Fax: ⫹ 386(0)61-218-567; e-mail: [email protected]

distribution function m X



s wl exp ⫺ F …s† ˆ 1 ⫺ u l lˆ1

bl !

…1†

is suitable for the load ranges s of stationary random processes. Constant m stands for the number of Weibull distributions, wl are used as weighting factors, b l and u l represent Weibull parameters. By using two parameter Weibull distribution, it is possible to approximate well almost all known distributions. Most probably that is why this distribution is used so widely for loading spectra modelling. Since the shape of a loading spectrum is often composed of more than one basic shape, a logical step forward is to introduce the multi-modal Weibull distribution as the most appropriate distribution for loading spectra modelling. The proposed distribution might seem to be rather complicated because of a larger number of unknown parameters. This is only partially true. However, when the design spectrum has to be defined, the proposed distribution turns out to be more efficient and straightforward than any of the currently available distributions [3]. Based only on the knowledge about probability of occurrence for the spectrum, a reliable life determination could be carried out [1]. Since the duration of the field measurements is limited due to technical and economical reasons, the results of the measurements must be properly extrapolated.

0951-8320/99/$ - see front matter 䉷 1999 Elsevier Science Ltd. All rights reserved. PII: S0951-832 0(98)00085-4

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M. Nagode, M. Fajdiga / Reliability Engineering and System Safety 64 (1999) 383–389

A new procedure of loading spectra extrapolation and its scatter prediction was developed [4]. It is based on a general multi-modal p.d.f. Eq. (1) as well as on the theory of extreme values. The scatter of the loading spectrum is in that case completely determined by two conditional p.d.f.’s. By the conditional p.d.f. of load ranges

factor A takes the value corresponding to level a. To make further steps as clear as possible, only three factors A, B, C are going to be taken into account. If factors A, B, C represent statistically independent random variables, the probability of a particular factor level combination AaBbCc is given by

ÿ
f s兩n ˆ

pAa Bb Cc ˆ pAa pBb pCc ;

N! F …s†N⫺n …1 ⫺ F …s††n⫺1 f …s†; …N ⫺ n†!…n ⫺ 1†! …n ˆ 1; …; N †

…2†

and by the conditional p.d.f. of the number of load cycles f(n|s). Conditional p.d.f. f(n|s) is Gaussian distribution with mean value m(s) and standard deviation s (s) given by m…s† ˆ 1 ⫹ …N ⫺ 1†…1 ⫺ F …s†† and p s…s† ˆ …N ⫺ 1†…1 ⫺ F…s††F…s†

…3†

Variable n stands for the number of load cycles, whereas N stand for the size of the loading block. Until now the appropriateness of F(s), f(s|n) and f(n|s) was verified only for the case of fixed operating conditions. The scope of the article is to prove that the same distribution models hold also for the cases of variable operating conditions.

During its exploitation a structural component is subjected to variable operating conditions that can generally be characterized by a certain number of significant factors, each observed at a definite number of levels [5, 6]. For example, it is known that the loads acting upon the structure of a fork-lift are significantly influenced by manoeuvres like lifting and leaning the load or driving direction, driving surface, driving speed and the mass of additional load. Let the significant factors be denoted by capital letters A, B, C,… Each of them can be treated as a discrete or continuous random variable, distributed according to F(A), F(B), F(C),… Theoretically speaking, the total number of possible operating conditions for the cases when one or more factors are of the continuos-type tends to infinity. Since only a finite number of measurements can be carried out, continuous as well as discrete factors should be observed only at a definite number of levels nA, nB, nC, … Therefore the total number of possible operating conditions equals the number of factor level combinations nT ˆ nA nB nC … Let indices a, b, c, … denote the successive level number that belongs to Aa (a ˆ 1,...,nA), Bb (b ˆ 1,...,nB), Cc (c ˆ 1, …,nC), … It is always possible to assign certain non-negative probability to every significant factor level in such a way that p Aa

nA
X ⱖ 0; a ˆ 1; …; nA ; pAa ˆ 1

b ˆ 1; …; nB ;

c ˆ 1; …; nC †

…5†

where nC nA X nB X X

pAa Bb Cc ˆ 1:

aˆ1 bˆ1 cˆ1

If variables are statistically dependent, each pAa Bb Cc has to be stated on the basis of the predicted usage of a component. The expected service life is usually expressed by the number of load cycles N and not by time t. Therefore the relation between the time belonging to particular factor level combination and the number of load cycles has to be defined. Under stationary random loading, the expected number of cycles in time t is given by [7] E‰N…t†Š ˆ np t;

…6†

where N(t) is the total number of cycles in time t and np is the peak rate. The total number of load cycles of design loading spectrum N can be expressed as

2. Verification of distribution functions

ÿ

…a ˆ 1; …; nA ;

…4†

aˆ1

Eq (4) holds. Variable pAa stands for the probability that

Nˆ

nC nA X nB X X

…7†

NAa Bb Cc

aˆ1 bˆ1 cˆ1

N and NAa Bb Cc will be referred to as the size of the loading block of design and partial loading spectra respectively. If in Eq. (7) the sizes of loading blocks are replaced by Eq. (6), the equation can be rewritten as vp t ˆ

nC nA X nB X X

vAa Bb Cc tAa Bb Cc :

…8†

aˆ1 bˆ1 cˆ1

Substituting vAa Bb Cc in Eq. (8) with tAa Bb Cc ˆ pAa Bb Cc t; …a ˆ 1; …; nA ;

b ˆ 1; …; nB ;

c ˆ 1; …; nC †

…9†

yields the peak rate of design loading spectrum vp vp ˆ

nC nA X nB X X

pAa Bb Cc vAa Bb Cc :

…10†

aˆ1 bˆ1 cˆ1

Peak rate vAa Bb Cc is given by vAa Bb Cc ˆ

NAⴱ a Bb Cc ; tAⴱ a Bb Cc

…a ˆ 1; …; nA ;

…11† b ˆ 1; …; nB ;

c ˆ 1; …; nC †

where tAⴱ a Bb Cc and NAⴱ a Bb Cc stand for known values. The former represents the return period of the measured load

M. Nagode, M. Fajdiga / Reliability Engineering and System Safety 64 (1999) 383–389

m ˆ nT mmax)

time history sample, whereas the latter stands for the related size of the loading block. The relation between relative frequencies in time domain pAa Bb Cc and relative frequencies of the number of load cycles p 0Aa Bb Cc can now be expressed as p 0Aa Bb Cc

NAa Bb Cc vA B C ˆ a b c pAa Bb Cc ; ˆ N vp

…a ˆ 1; …; nA ;

b ˆ 1; …; nB ;

 bl ! s wl exp ⫺ ; F…s† ˆ 1 ⫺ u l lˆ1 m X

nC nA X nB X X

…12†

c ˆ 1; …; nC †

p 0Aa Bb Cc FAa Bb Cc …s†

…13†

p 0Aa Bb Cc

aˆ1 bˆ1 cˆ1

 1⫺

X

mAa Bb Cc

wAa Bb Cc l exp ⫺

lˆ1

!bA B

a b Cc l

s

!! :

uAa Bb Cc l

Further, the above equation can be rewritten as F…s† ˆ 1 ⫺

nC nA X nB X X

p 0Aa Bb Cc

aˆ1 bˆ1 cˆ1

X

mAa Bb Cc

wAa Bb Cc l exp ⫺

lˆ1

s

!bA

a Bb Cc l

! :

uAa Bb Cc l

…14† If the number of Weibull distributions mAa Bb Cc is replaced by mmax ˆ max{mAa Bb Cc ; (a ˆ 1, …,nA; b ˆ 1, …,nB; c ˆ 1, …,nC)} and value wAa Bb Cc l ˆ 0 is added to the weighting factors of which l ⬎ mAa Bb Cc is true, Eq. (14) yields F…s† ˆ 1 ⫺

nC m nA X nB X max X X

p 0Aa Bb Cc wAa Bb Cc l

aˆ1 bˆ1 cˆ1 lˆ1

exp ⫺

s

!bA B

a b Cc l

!

uAa Bb Cc l

where the products of p 0Aa Bb Cc wAa Bb Cc l represent weighting factors w 0Aa Bb Cc l ˆ p 0Aa Bb Cc wAa Bb Cc l ; …a ˆ 1; …; nA ; b ˆ 1; …; nB ; c ˆ 1; …; nC ; l ˆ 1; …; mmax † …15† As nA, nB, nC and mmax are constant values that are independent of indices a, b, c and l, the sum over four indices can be transformed into the sum over single index l (l ˆ 1, …,

…16†

The basic characteristics of the existing method for design spectrum prediction, used mainly for light-weight design in automotive industry [8], are as follows:

If distributions FAa Bb Cc …s† are substituted with Eq. (1), we get nC nA X nB X X

wl ˆ 1:

lˆ1

3. Possible approaches for design spectrum determination

aˆ1 bˆ1 cˆ1

F…s† ˆ

m X

As Eq. (16) is identical to Eq. (1), random variable s is distributed according to the multi-modal Weibull distribution function also in the case of variable operating conditions. The same expression Eq. (16) would be obtained, if instead of for three, the above proof was carried out for any number of significant factors.

Assuming that distribution functions of load ranges FAa Bb Cc …s† as well as relative frequencies p 0Aa Bb Cc are known for all possible operating conditions, the distribution function of the design loading spectrum is given by F…s† ˆ

385

• Service life depends primarily on the loading conditions in service. They can be characterized by a design spectrum. The main parameters of the spectrum are usage, structural behaviour and operating conditions. To make the application of the method possible, these parameters have to be known in advance. • For a reliable evaluation of service fatigue life the probability of occurrence of the design spectrum has to be defined. • To define the design spectrum, service measurements are needed to determine either the data for different operating conditions or the customer usage as well. • The values originating from different loading conditions are separated by filtering. Measured load time histories are extrapolated and partial spectra are worked out. • The design spectrum is assembled from partial loading spectra. Instead of currently available distribution functions a multi-modal Weibull distribution can be used to extrapolate partial spectra [3]. In this case unknown constants wl, b l and u l belonging to FAa Bb Cc …s† for all factor level combinations should be defined. Consequently the distribution function of the design spectrum F(s) can be calculated by using Eq. (13). The rise of the number of factor level combinations nT results in a sharp increase of partial spectra. Therefore defining F(s) is extensive and time-consuming, irrespective of the chosen distribution function. A much more efficient way to obtain the design spectrum follows from expression Eq. (16). It consists of the next steps: • By carrying out measurements (simulations) loading spectra for all possible factor level combinations are obtained. • From record lengths tAⴱ a Bb Cc and from the calculated sizes of loading blocks NAⴱ a Bb Cc peak rates vAa Bb Cc and vp are evaluated. Relative frequencies in time domain pAa Bb Cc

386

M. Nagode, M. Fajdiga / Reliability Engineering and System Safety 64 (1999) 383–389

are transformed into relative frequencies of the number of load cycles, using Eq. (12). • Relative or relative cumulative frequencies of a design spectrum are given by

Table 1 Significant factors and the corresponding number of levels Manoeuvres Driving straight on Driving backwards Left turn Right turn

A1 A2 A3 A4 S

Driving speed (km/h) 0⫼7 7 ⫼ 14 14 ⫼ 21 Additional load (kg) 0.0 1260.7 2524.7

B1 B2 B3 S C1 C2 C3 S

pAa

NAⴱ a

nAa

p 0Aa

0.62 0.22 0.08 0.08 1.00 pBb 0.10 0.65 0.25 1.00 pCc 0.45 0.32 0.23 1.00

1553.0 1347.8 1394.0 1557.4 5852.3 NBⴱ b 1270.2 1607.4 2974.6 5852.3 NCⴱ c 2814.9 1507.7 1529.7 5852.3

16.98 16.36 15.66 17.79

0.627 0.214 0.075 0.085 1.000 p 0Bb 0.062 0.398 0.540 1.000 p 0Cc 0.637 0.226 0.137 1.000

nBb 10.36 10.29 36.33 v Cc 23.79 11.88 9.99

f 0i ˆ

nC nA X nB X X

p 0Aa Bb Cc f 0Aa Bb Cc i ;

aˆ1 bˆ1 cˆ1

F 0i

ˆ

nC nA X nB X X

…17† p 0Aa Bb Cc F 0Aa Bb Cc i

;

…i ˆ 1; …; k†

aˆ1 bˆ1 cˆ1

where k denotes the number of classes. • Finally, unknown constants wl, b l and u l, are calculated for a single histogram by the procedure dealt with in reference 2. It has been proved that in this case design spectrum determination is much easier and quicker, for it is only the

Table 2 Transformation of relative frequencies into relative frequencies of the number of load cycles a b c pAa Bb Cc

NAⴱ a Bb Cc

nAa Bb Cc

p 0Aa Bb Cc

a b c pAa Bb Cc

NAⴱ a Bb Cc

nAa Bb Cc

p 0Aa Bb Cc

a b

c pAa Bb Cc

NAⴱ a Bb Cc

n

1 1 1 2 2 2 3 3 3 4 4 4

102.2 70.9 807.2 150.7 101.5 630.3 130.7 128.9 219.5 128.7 165.7 178.8

10.22 7.09 80.72 15.07 10.15 63.03 13.07 12.89 21.95 12.87 16.57 17.88

0.017 0.076 0.335 0.009 0.039 0.093 0.003 0.018 0.012 0.003 0.023 0.010

1 1 1 2 2 2 3 3 3 4 4 4

137.3 81.8 155.0 56.0 181.3 42.3 41.3 151.3 226.0 123.3 120.1 191.7

13.73 8.18 15.50 5.60 18.13 4.23 4.13 15.13 22.60 12.33 12.01 19.17

0.016 0.063 0.046 0.002 0.049 0.004 0.001 0.015 0.009 0.002 0.012 0.007

1 1 1 2 2 2 3 3 3 4 4 4

3 3 3 3 3 3 3 3 3 3 3 3

50.8 110.8 37.0 53.9 48.7 83.1 143.8 150.8 201.7 151.5 295.6 202.0 5852.3

5.08 11.08 3.70 5.39 4.87 8.31 14.38 15.08 20.17 15.15 29.56 20.20

1 2 3 1 2 3 1 2 3 1 2 3

1 1 1 1 1 1 1 1 1 1 1 1

0.028 0.181 0.070 0.010 0.064 0.025 0.004 0.023 0.009 0.004 0.023 0.009

1 2 3 1 2 3 1 2 3 1 2 3

2 2 2 2 2 2 2 2 2 2 2 2

0.020 0.129 0.050 0.007 0.046 0.018 0.003 0.017 0.006 0.003 0.017 0.006

1 2 3 1 2 3 1 2 3 1 2 3 S

Fig. 1. Moment representing vertical force acting on the front left wheel ZL1.

0.014 0.093 0.036 0.005 0.033 0.013 0.002 0.012 0.005 0.002 0.012 0.005 1.000

Aa Bb Cc

p 0Aa Bb Cc 0.004 0.061 0.008 0.002 0.010 0.006 0.002 0.011 0.006 0.002 0.021 0.006 1.000

M. Nagode, M. Fajdiga / Reliability Engineering and System Safety 64 (1999) 383–389

387

Fig. 2. Moment representing vertical force acting on the rear left wheel ZL2.

distribution of a single, representative design spectrum that has to be defined. The p.d.f.’s of partial spectra are not required. It turns out that the number of Weibull distributions m Ⰶ nT mmax, too. Thus the analysis can take into account a much larger number of different operating conditions instead of including just some of the most inconvenient ones. A practical example confirming the preceding mathematical findings has been thoroughly discussed in reference [3].

4. Application of design spectrum determination During its operation the structure of a fork-lift is subjected to different operating conditions that can be characterized by a certain number of significant factors, as mentioned before. The significance of each individual factor can be tested with experiment design methods [6]. Let’s assume that the loads acting upon the structure of a forklift are significantly influenced by manoeuvres (factor A),

Fig. 3. Side force acting on the front left wheel YL1.

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M. Nagode, M. Fajdiga / Reliability Engineering and System Safety 64 (1999) 383–389

Fig. 4. Force acting on the tilting hydraulic cylinder ZVNH.

driving speed (factor B) and by additional load (factor C). Tables 1 and 2 show the transformation of relative frequencies in a time domain into relative frequencies of the number of load cycles. Record length tAⴱ a Bb Cc ˆ 10 s for all factor level combinations, whereas peak rate np ˆ 16:8. Relative frequencies pAa Bb Cc were calculated by Eq. (5). In the next step a design spectrum was composed of measured loading spectra, using Eq. (17). As the extrapolation of design spectrum as well as the scatter of extrapolated loading spectrum prediction are only possible if the distribution function of load ranges F(s) is known, unknown constants wl, b l and u l, have finally been calculated. Figures 1–4 depict the p.d.f.’s of load ranges f(s) belonging to different load sources as well as four different examples of extrapolated loading spectra with confidence limits. H0 and H0E stand for the size of the loading block of measured and extrapolated loading spectrum respectively. It is evident that measured relative frequencies f 0i and f(s) agree very well. The number of Weibull distributions m varies according to the measuring point, but m Ⰶ nT mmax. Since the distribution function of load ranges F(s) remains a multi-modal Weibull distribution also in the case of variable operating conditions, conditional p.d.f. of load ranges f(s|n) and conditional p.d.f. of the number of load cycles f(n|s) should remain unchanged, too.

5. Conclusion It was proved that the design spectrum is also distributed with the same distribution if partial spectra are distributed according to the multi-modal Weibull distribution. The only difference between the two distributions is in the values of

unknown constants wl, b l and u l. As these parameters have to be calculated only for a single spectrum, the analysis can take into account a much larger number of different operating conditions instead of including just some of the most inconvenient ones. Since the conditional p.d.f. of load ranges f(s|n) and of the number of load cycles f(n|s) depend solely on F(s), conditional distributions remain unchanged in the case of variable operating conditions, too. The parameter estimation for the multi-modal Weibull distribution is not as straightforward as the parameter estimation for distributions used presently. The same is true also for the scatter of loading spectra prediction. However, with the proposed procedure parameters and scatter may be worked out just for a single spectrum. Consequently, the design spectrum prediction is much faster and simpler, which is most probably the major advantage of the proposed procedure.

References [1] Grubisˇic´ V. Determination of load spectra for design and testing. Int. J. of Vehicle Design 1994;15(1/2):8–26. [2] Nagode M, Fajdiga M. A general multi-modal probability density function suitable for the rainflow ranges of stationary random processes. Int. J. of Fatigue 1998;20(3):211–223. [3] Nagode, M. & Fajdiga, M., On the new method of the loading spectra extrapolation and its scatter prediction, Fatigue Design 1998, Espoo, Finland, vol. 2 Technical Research Centre of Finland (VTT) (1998), pp. 385–391. [4] Nagode M, Fajdiga M. On a new method for prediction of the scatter of loading spectra. Int. J. of Fatigue 1998;20(4):271–277. [5] Nagode, M., Experiment design for operating strength. M.Sc. Thesis, University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, 1994.

M. Nagode, M. Fajdiga / Reliability Engineering and System Safety 64 (1999) 383–389 [6] Taguchi, G., System of experimental design vol. 1 & 2, American Supplier Institute Inc., Dearborn, Michigan, 1987. [7] Zhao W, Baker M J. On the probability density function of rainflow stress range for stationary Gaussian processes. Int. J. of Fatigue 1992;14(2):121–135.

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