In automotive industry, temporal, financial and human constraints impose to improve the design process of new vehicles. The group PSA PEUGEOT CITROËN ...
ORAL REFERENCE : FT122
MULTI-INPUT EQUIVALENT FATIGUE LOADINGS 1−2
Gwenaëlle Genet, 1 Pär Johannesson, 2 Mac Lan Nguyen-Tajan.
1 FRAUNHOFER-CHALMERS
Center for industrial Mathematics, Science Park, SE-412 88 GÖTEBORG, SWEDEN, 2 PSA PEUGEOT CITROËN, Route de Gisy, F-78943 VÉLIZY, FRANCE.
ABSTRACT In automotive industry, temporal, financial and human constraints impose to improve the design process of new vehicles. The group PSA PEUGEOT CITROËN uses stress strength reliability approach to reach optimal design for mechanical components of cars. Driver’s behaviors or types of roads influences the reliability of components. The time history of random oscillating multidimensional forces contains this information. The equivalent fatigue approach is a method for transforming complicated measured forces into simpler loadings, equivalent in terms of damage. The transformation should be performed without explicit information about the structure. The aim is to present an extension of the equivalent fatigue to multidimensional loadings, generating multi-input equivalent loading in terms of damage. We use the principle of equal damage to estimate suitable parameters of the multi-input equivalent loadings. Narrow band Gaussian and sinusoidal multi input equivalent loads are described. KEYWORDS Multiaxial fatigue, Morel, narrow band Gaussian, equivalent fatigue, SN curve, Basquin. INTRODUCTION The equivalent fatigue approach is a method for transforming variable amplitude measured forces into simpler loadings, equivalent in terms of damage. The transformation should be performed without information about the structure. The industrial target of this equivalent multi input loadings is triple: Its use will intervene in the computations for
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dimensioning metallic structures, in the specifications of their tests trials, and finally on the characterization of the severity of customers’ behaviors and markets. The main ideas of the method of multi-input equivalent fatigue loadings will first be recalled. Damage is derived from the stress tensors. However, in the context of equivalent fatigue approach, the stress tensors are unknown. Hence, it is necessary to estimate the damage directly from the loadings. This will be performed into two different steps: the evaluation of the stress tensors from multi-input loadings and the estimation of the damage from stress tensors. In this last task, the Morel’s life prediction method for high cycle multiaxial fatigue of metallic materials will be used. In the final expression of the damage, some parameters related to the geometry will appear. We will attempt to characterize the structures for which it is possible to evaluate them. Finally, we will use the equivalence of damage to define suitable parameters for the multiinput equivalent loadings. We describe different kinds of equivalent loads, such as narrow band Gaussian processes or sinusoidal multi-input signals. Several applications will be developed. EVALUATION OF THE DAMAGE FROM MULTI-INPUT FORCES During the whole life of car components, the behavior is assumed to be elastic and quasi static. Hence, the macroscopic stress tensor is a linear combination of the loadings. Moreover, in any structures undergoing multiaxial loadings, cracks initiation appears more likely at the critical points of the structure. The durability of the components completely depends on the behavior of these critical points. It has been frequently observed that, on components of a car, the principal directions of stress tensors at the critical points don’t rotate in time : the stress tensors are proportional or even unidirectional. That is why we have restricted the construction of the multi-input EFL to these cases. The framework of the equivalence fatigue approach is the high cycle fatigue for finite life prediction. High cycle fatigue criteria for infinite life can not be used in the method, because it doesn’t allow us to evaluate damage. Moreover, the predicted damage has to be linked to variables that can be expressed from forces. In the case of multi-input equivalent fatigue loadings, different kind of sequences of forces are involved (variable amplitude loadings and constant amplitude loadings). Hence we need to predict the life from variable amplitude and constant amplitude loadings. The case of unidirectional stress tensor has been detailed in the literature, concerning the one-input equivalent fatigue loadings, [1], [2], [3]. The Basquin’s model was used. In case of unidirectional stress tensors, applied to multi-input equivalent fatigue loadings, Basquin’s model is still valid. However, for proportional multidimensional stress tensors, the computation of the damage using Basquin’s model is not appropriate anymore. A
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multiaxial fatigue criterion for finite life prediction would be more suitable. From experimental observations, we observed that cracks nucleate and grow on specific planes of some grains, in the microscopic scale, leading to the deterioration of the material and initiation of cracks. Dang Van is the precursor of this approach and proposed a well known high cycle fatigue criteria for infinite life [4]. Papadopoulos also provided a criterion based on these concepts [5]. Morel’s work, inspired by the Dang Van’s approach, provided a criterion in high cycle multiaxial fatigue for finite life, [6], [7]. From the Morel’s model of life prediction, we have deduced the damage induced by n variable amplitude forces {F1 (t), . . . , Fn (t)}(for more details, see [8]). Again, this expression of the damage is valid for proportional or unidirectional stress tensors : Dseq (Ac ) =
2 nu 2(Fu∗rf c (Ac ) − T (Ac ))+ + Res(F ∗rf c (Ac ))]. [ qT (Ac ) u
with F ∗ (Ac , t) =
n i=1
ai (Ac ) Fi (t)
n
a2i (Ac ) = 1 C(Ac ) > 0, i=1
.
(1)
(2)
Let’s call the set of ai with respect of conditions of Eq. (2) ∆(Ac ). ∆(Ac ) = {a1 (Ac ), a2 (Ac ), . . . , an (Ac )}. Fu∗rf c (Ac ) is the range of the uth rainflow cycle, nu its number of occurrence. Parameters ∆ and T depend on the material and on the geometry. They are constant whatever the loadings applied on the geometry considered. The parameter q is a constant dependant of the material, and can be deduced from an experimental Wöhler curve. A part of the residual, called Res(F ∗rf c (Ac )) of F ∗rf c induces damage. The details about its treatment won’t be detailed in this paper. EVALUATION OF CONSTANTS RELATED TO THE GEOMETRY If the geometry and the critical point Ac is known, ∆(Ac ) and T (Ac ) can be computed. The damage can be evaluated for this point. However, in our case, the structure is unknown. In order to evaluate the damage, we need to evaluate T and ∆. The problem of finding a suitable threshold T whatever the geometry, seems to be impossible to solve without additional assumptions on components we are working on. In any number of industrial cases, structures are designed in order to fulfill reliability requirements, imposed by the manufacturer. These requirements can have different forms. In this case, we consider that the whole sequence of forces that the structures has to bear without reaching ruin, is known. This sequence can be expressed as forces measured and stored during test tracks. The number of laps the structure has to sustain is
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predefined by the manufacturer. Hence, we know that structures fulfilling the reliability requirements have a damage reaching one at the critical point, when those forces, called Fdl,{1} , . . . , Fdl,{n} , are applied. Definition: A structure is optimally designed if after applying the sequences Fdl,{1} , . . . , Fdl,{n} c , of forces, at its critical point A c ) = 1. Ddl (A The aim is to characterize all the different geometries and critical points, that are optimally c on an optimal structure is defined by its threshold T (A c ) and designed. A critical point A c ). For each of these points, Eq. (3) is fulfilled: linear combination ∆(A 2 ∗rf c c ))+ + Res(F ∗rf c (A c ))] = 1. nu 2(Fdl,u (Ac ) − T (A [ dl qT (Ac )
(3)
u
The parameter Fdl∗ is the linear combinations of the sequence Fdl,{1} , . . . , Fdl,{n} of loadings. In order to characterize all the optimal structures, we need to solve this equation for all the c ). Hence we are able to characterize the critical points, different linear combinations ∆(A through ∆ and T , of all the different optimal structures. 4
4
1.5
x 10
1.7
x 10
1
1.6
0.5
F
{dl},1
T(0)=1.556.104 N
0 −0.5
1.5
−1 −1.5 0
0.5
1
1.5
2.5
3
3.5
1.4
4 4
x 10
T
4
1.5
2
TIME
x 10
1.3
1 0.5
F{dl},2
1.2
0 −0.5
Tmin(55) = 1.085 104 N
1.1
−1 −1.5 0
0.5
1
1.5
2
TIME
2.5
3
3.5
4 4
1 0
20
40
60
80
x 10
100
γ [deg]
120
140
160
180
Figure 1: Evaluation of the threshold T
In Fig. (1), an example of characterization of optimal geometries is proposed. The first plot illustrates a part of the sequences of the forces Fdl,{1} , Fdl,{2} . Let’s consider structure reaching ruins when 180 times these sequences are applied on them. The aim is to characterize the structures, optimally designed for these sequences. In the bidimensional case, we have chosen to express the linear combination by a parameter γ. From Eq. (2): c ) = {a1 , a2 } = {cos(γ), sin(γ)}. ∆(A Hence,
Fdl∗ (γ) = cos(γ) Fdl,{1} + sin(γ) Fdl,{2} .
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c ), T (A c )) represent all the different optimal structures and critThe different couples (γ(A c . They fulfill Eq. (3): ical points A 180
2 ∗rf c c ))+ + Res(F ∗rf c (A c ))] = 1. nu 2(Fdl,u (Ac ) − T (A [ dl c ) qT (A u
The parameter q has been evaluated from the type of material we consider. In the second c ), in respect to γ(A c ) are represented. The plot of Fig. (1), the different threshold T (A calculation has been done for γ = [0, 5, 10, . . . , 175] degrees. For this calculation, we need to evaluate q. In case of steels, q 105 .
CONSTRUCTION OF EQUIVALENT FATIGUE LOADINGS After evaluating the damage from multi-input loadings using Morel’s model, we have fixed the property that the equivalent multi-input loadings has to fulfill in order to be equivalent. This equivalence of damage enables us to define suitable parameters of the multi-input equivalent loadings. We can consider different models of equivalent loadings, either deterministic or probabilistic representation. Let’s consider θ, belonging to the parameter space Θ, the vector containing all the parameters needed to characterize the multi-input equivalent loadings. Damages D and De are induced by the measurements and the equivalent loadings respectively. Using the least square method, the vector θ defines the multi-input equivalent loadings if it satisfies: m 1 eq {i} {i} (4) (D (θ, (Ac )) − D(Ac ))2 , θ = arg min θ∈Θ m i=1 {i} where Ac represents the critical points of the optimal structures. From Eq. (1),we need to define amplitudes of rainflow cycles of linear combinations F ∗e and mean number of occurrences, from parameters describing the equivalent forces, belonging to θ. Evaluation of rainflow intensity has been provided for random processes like Markov chain [9] and [10], or Gaussian loadings [11]. For deterministic loadings, like sinusoidal loadings, the rainflow intensity, can be easily provided. For more details, see [8]. In the followings, we attempt to evaluate damage from sinusoidal and Gaussian loadings, equivalent in terms of damage to measurements, called {F1 (t), . . . , F2 (t)}. The multi-input equivalent loadings are called {F1e (t), . . . , Fne (t)}.
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Sinusoidal multi-input equivalent loading We choose to estimate a multidimensional equivalent loading: Fie (t) = Bi cos(ωi t + φi ) with 0 ≤ t ≤ Tm . The equivalent loads are chosen to have the same fixed number of cycles N0 representing one design life. This means that: ωi = ω = 2πN0
i = 1, . . . , n.
We need to find the vector of parameters θs in order to define the equivalent loadings. The set of parameters θs is belonging to the parameter space Θs : θs ∈ Θs = {B1 > 0, ..., Bn > 0, 0 ≤ φ1 < 2π, ..., 0 ≤ φn < 2π}. As explained in [8], we have deduced the expression of the damage : e Dseq (Ac ) =
2 2N0 (2B ∗ (Ac ) − T )+ , qT
with ∗
B (Ac ) =
n
ai (Ac ) Bi
i=1
This expression of damage from sinusoidal loads is available for proportional stress tensors. We now have to define the vector of parameters θs in order to get the same damage at c . Hence, all ∆, have to be taken into account. Using the least square the critical points A method, the suitable vector θs have to fulfill the condition explained in Eq. (4).
Gaussian equivalent multi-input loadings Let’s consider n different independent Gaussian processes {X1e (t)}, . . . , {Xne (t)} and Gaussian processes {F1e (t)}, . . . , {Fne (t)}, equivalent in terms of damage to measurements. The different statistical parameters of the equivalent processes can be expressed by those of independent Gaussian ones. The Gaussian equivalent loadings are also chosen to be narrow band processes. Hence, the Rayleigh approximation is providing an estimation of the intensity of rainflow cycles for narrow band stationary Gaussian processes. The Rice’s formula allows us to determine the number of level crossing of a Gaussian process [11],[12]. As far as the processes {F1e (t)},...,{Fne (t)} are the projection of the independent processes {X1 (t)},..,{Xn (t)}, they are linked by some linear combinations deduced by the change of axis system. If we write: n e∗ ai (Ac ) Fie (t) F (Ac , t) = i
then F (Ac ) = e∗
n
a∗i (Ac ) Xie (t).
i
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The final expression of the damage can be then deduced (see [8]): √ 4 ∗ T e Dseq (∆) = σ (∆) N0 2π 1 − Φ qT σ ∗ (∆) with Φ, the standard normal cumulative distribution function, and
n σ∗ = (a∗i σi )2 . i=1
The parameter N0 is the fixed number of level crossing of the mean, which can be approximately the number of Rainflow cycles. The vector of parameters to find is called θG and is belonging to ΘG : θG ∈ ΘG = {0 ≤ β1 < 2π, ..., 0 ≤ βn < 2π, σ1 > 0, ..., σn > 0} . where the different angles β1 , ..., βn are defining the change of axis system in the ndimensions space. The suitable vector θG is fulfilling the minimization of Eq. (4). APPLICATIONS An example of bidimensional sinusoidal equivalent fatigue loadings is presented. The thresholds have been previously computed in fig 1. However, the fitting of the damage leads to great difference between the damage induced by the equivalent loadings and the measurements, Fig. (2). 8000 1.8 D De
6000 1.6
4000
2000 e
F2 0
1.4 1.2
Amplitude Fe1=7815 D
Amplitude Fe=6474 2
1 0.8
−2000
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−4000
0.2
−6000 0 0
−8000 −8000 −6000 −4000 −2000
0
2000
4000
6000
8000
50
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350
400
γ [deg]
e
F1
Figure 2: Sinusoidal equivalent fatigue loadings and fitting of De over the optimal structures
In order to provide better results, we extended the models of bidimensional sinusoidal loadings with two different blocks. Each block is composed by N20 cycles. The results are represented in Fig. (3).
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8000 1.8
6000
1.6
4000
1.4
Amplitude Fe1 7925 N Amplitude Fe2=6371 N
2000 e
24%
1.2 1
D
0
F2
0.8
e
Amplitude F1 = 7284 N Amplitude Fe2 = 6807 N
−2000
D=0.77
0.6 37%
−4000
0.4
−6000
0.2 0 0
−8000 −8000 −6000 −4000 −2000
0
2000
4000
6000
50
100
8000
150
200
250
300
350
γ [deg]
e F 1
Figure 3: Sinusoidal equivalent fatigue loadings with two sinusoids (left) and fitting of De over the optimal structure (right)
The same work has been done, using the Basquin model of evaluation of the damage, in Fig. (4). 4
4
1
x 10
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
x 10
e{B} Amplitude F1 = 8625 Amplitude Fe{B} = 7463 2
Fe{B} 2
e{B}
Fe{B} 2
Amplitude F1 = 8350 Amplitude Fe{B} = 7780
0
0
2
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1 −1
−0.5
Fe{B} 1
0
0.5
1 4
x 10
−1 −1
Amplitude Fe{B} = 6656 1 Amplitude Fe{B} = 7528 2
−0.5
0
Fe{B} 1
0.5
1 4
x 10
Figure 4: Sinusoidal equivalent fatigue loadings with one sinusoid(left) and two sinusoids (right)
Comparing Fig. (3) and Fig. (4), we deduce that the evaluation of the sinusoidal equivalent fatigue loadings is very dependant of the criterion we use. Moreover, in order to illustrate the Gaussian equivalent loadings, an application of Gaussian equivalent fatigue loadings is proposed in the Fig. (5).
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σ(Fe) = 456.3
4000
2000
2000
F1
Fe
0
−2000
−1000 2
4
6
8
10
12
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16
−2000 0
18 4
TIME
x 10
3 TIME
4
3 TIME
4
5
6 4
x 10
e
1000
F2
2000
0
0 −2000 0
2
2000
4000
2
1 e σ(F2) = 478.3
6000
F
1000
1
0
−4000 0
1
−1000 2
4
6
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10
12
14
16
18 4
x 10
TIME
−2000 0
1
2
5
6 4
x 10
Figure 5: Measurements (left) and Gaussian equivalent fatigue loadings (right)
In this case, the threshold has been considered as constant: T = 2800N. The measurements are representing longitudinal and transversal loadings. CONCLUSION The evaluation of the damage from the forces, under hypothesis of proportional stress tensors and quasi static and elastic structures has been recalled. In order to evaluate constants related to the geometry, we needed to restrict the study to structures for which the forces representing their design life, are known. Information about the reliability requirements are necessary in order to evaluate constants related to the geometries of optimally structures. Some applications has been presented. In case of sinusoidal loading, a mix of different blocks of sinusoids provides better fitting of damage, over the linear combinations. The use of two different models of life prediction leads to different equivalent loadings. An application to Gaussian equivalent loadings has been presented. For sinusoidal and Gaussian loads, the extension to more than two input loads is also possible. ACKNOWLEDGEMENT Thanks go to my supervisors from PSA PEUGEOT CITROËN, Dr. Ida Raoult, and also Dr. Jean-Jacques Thomas, and my main supervisor from CHALMERS, Professoer Jacques de Maré, and finally Professor Igor Rychlik (LUND University-CHALMERS) for their supports and valuable comments throughout this work.
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References [1] P. Johannesson, J.-J. Thomas, and J. de Maré. Extrapolation and scatter of test track measurements. Fatigue 2002, Stockholm, Sweden, June 2-6, 2002. [2] A. Bignonnet. Fatigue desing in automotive industry. High Cycle Metal Fatigue in the Context of Mechanical Design, 1997. [3] J.-J Thomas, G. Perroud, A. Bignonnet, and D. Monnet. Fatigue design and reliability in the automotive industry. In G.Marquis and J.Solin, editors. Fatigue Design and Reliability, ESIS publication., 23:1, Elsevier, 1999. [4] K. Dang Van. Sur la résistance à la fatigue des matériaux. Sciences et Techniques de l’Armement, Mémorial de l’Artillerie française, 1973. [5] I.V. Papadopoulos. Fatigue polycyclique des métaux: une nouvelle approche. PhD thesis, École Nationale des Ponts et Chaussées, Paris, 1987. [6] F. Morel. A fatigue life prediction method based on a mesoscopic approach in constant amplitude multiaxial loading. Fatigue & Fracture of Engeneering Materials & Structures, 21:241, 1998. [7] F. Morel. A critical plane approach for life prediction of high cycle fatigue under multiaxial variable amplitude loading. Int. Journal of Fatigue, 22:101, 2000. [8] G. Genet. An approach to multi-input equivalent fatigue loadings. American Society of Mechanical Engineering, 2005. [9] N. W. M. Bishop and F. Sherrat. A theoritical solution for the estimation of. [10] M. Olagnon. Practical computation of statistical properties of rainflow counts. Internation Journal of Fatigue, 16:306, 1994. [11] S.-O Rice. The mathematical analysis of random noise, i and ii. Bell Syst. Technical Journal, 1944. [12] I. Rychlik. On the narrow banded approximation for expected fatigue. Probabilistic Engineering Mechanics, 1993.
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