on modeling the gear damage accumulation process

8/7/2019 On Modeling the Gear Damage Accumulation Process

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ON MODELING THE GEAR DAMAGE

ACCUMULATION PROCESS

К МОДЕЛИРОВАНИЮ ПРОЦЕССА АККУМУЛИРОВАНИЯРАЗРУШЕНИЙ В ЗАЦЕПЛЕНИЯХ

Lect. Dr. Eng. Mirica R.F.

1

, Prof. Dr. Eng. Dobre G.

1

, Dipl. Eng. Sandu C.

2

,Dipl. Eng. Dobre D 1

University “POLITEHNICA” of Bucharest 1, TMCP 2, [email protected], geo@ meca.omtr.pub.ro, [email protected]

Abstract : The paper discusses aspects regarding the modelling gear damage accumulation process useful for the gear reliability

distribution calculus. Because the real loading levels are found under and above the transition domain of the S-N (Wöhler) diagrams (closeby the endurance limit), it is useful to abandon the S-N family modelled by the Basquin function. The present paper proposes to use another

form of S-N curves, without a break at the knee. In this case, the classical accumulation theories (Schott or Haibach) cannot be applied. Asa result, a theory of damage accumulation is proposed, which considers the modification of the calculus stress amplitude depending on

accumulated damage. Finally, an application is analysed and a comparison with experimental results is made.KEYWORDS: GEAR, DAMAGE ACCUMULATION THEORY, S-N DIAGRAM

1. IntroductionThe paper discusses aspects regarding the modelling of gear

damage accumulation process used in the gear reliabilitydistribution calculus. In this paper the fatigue damage is only takeninto account from the multitude of damage modalities of the

toothed gears, which is evolutionary as the wear and produces thecontinuous growth of the failure probability. The teeth fatigue

rupture is considered in the standards for gear calculation methods(for example DIN 3990: 1987) as one of their most important

failure modes by normal operation.The estimation by calculus of the reliability distribution can bemade using a model of the damage accumulation. A correct

modelling of this process supposes on one hand a good understanding of the damage processes and this evolution in the

time and on the other hand the empirical description of the

influence of different factors as the load spectrum form, the loading level etc. In this context there are two research directions:a) a better approach of the fatigue process evolution

determining yet usually very complicated calculus methods;b) a better inclusion in calculus of the experimental results (for

example, the S-N (Wöhler) curve form without knee point, thedecreasing of the endurance limit alongside the damage

accumulation etc.).It is presented a proposal for the modelling the damageaccumulation process taking into account an important real

physical phenomenon of this process: modifying the damaging effect of cycles alongside the damage accumulation.

2. Damage accumulation process aspects

2.1. On accumulated damaging energyThe fatigue damage of metallic materials that can produce thefailure of mechanical components consists in the progressive

modification of mechanical and physical properties pursuant to asufficiently great number of loading cycles (damage

accumulation). This modification is determined by the initial material condition, the stress state nature, the stress history and the influence of the environment. It is reflected in Schott (1990):

a) microscopic material structure change and correlated withthis effect - the energy accumulation and the entropy

augmentation;b) hardness variation;

c) hysteresis loop variation;

d) internal damping change and other characteristics.By cyclical stress, the hysteresis loop surface is proportional to the

mechanical energy consumed by friction in that cycle, i.e. thevariable stress of the materials is accompanied by the internal

friction phenomenon. Hereby, a part of the mechanical oscillations

energy is transformed into heat and another part is included in thestructure of the material that is being deteriorated. Accepting that

the proportion of damaging accumulated energy into the total internal damping energy is maintained, it results that the variationof this accumulated energy may be correlated with the hysteresis

loop variation dependent upon the cycle number (figure 1).

2.2. On S-N (Wöhler) curve familyThe S-N (Wöhler) curve family establishes the relation between

stress σ and cycle number N having the failure probability r P as a

parameter:

( )r P ,f N σ = (1)

The S-N curve family is the model of the damage accumulationprocess for constant amplitude loading, in the sense that the curve

form indicates the increasing failure probability depending on thestress and the cycle numbers. Today the S-N curves or their family

represent a basis in defining numerous models of damageaccumulation process in the extended case of variable loading.

This important case will be discussed later in the paper.

2.3. On decreasing the endurance limit with the damageaccumulationThe representation from the figure 2 is used in the following considerations. The fatigue test is carried out under a stress at a

value grater than the endurance limit ( 0D0 σ σ > ) on a limited

cycle numbers - at a number of cycles exceeding the one

corresponding to the line of French ( F nn > ) - and being smaller

than the one corresponding to the S-N curve ( N n < ). The stress is

a)

ε ap

ε

σ a

σ

ε ap

(1)

104 n [cycles]2⋅ 104 3⋅ 104

5⋅ 10-4

10-3

(3)

(2)

b)

Figure 1. Hysteresis loop variation with the cycle number Schott (1990)

a) hysteresis loop; b) variation of plastic strength

σ a stress amplitude; ε ap - plastic strain



smaller than the endurance limit ( 0Dσ σ < ) but rupture may

occurs. The testing results were interpreted by a number of authors

(Gnilke, 1980; Haibach, 1989; Schott, 1990) as a decrease of the

endurance limit ( 0DD σ σ < ).

These authors did not consider correctly the physical phenomenonof the material internal friction. Really the material internal

friction variation determines the increase of the damaging effect of

the loading at lower amplitude σ mentioned above. The results of

fatigue bending tests of specimens by steel (SAE 1030), presented by Schott (1990) after Kommers, are given in figure 3. It is

observed that the decrease of the relative current endurance limit

is more significant when the pre-stress 0σ is higher. This fact may

be justified only by the existence of the material internal frictionincreasing with the preload.

The existence of the endurance domain of the S-N (Wöhler)diagram requires that in calculus the damage accumulation of thecycles with amplitudes in that domain is neglected. The problem is

additionally complicated by the necessity to consider thephenomenon of gradual decrease of endurance limit analysed

before (determined by pre-load at higher level as the endurancelimit). For this reason, in a great number of theories (hypothesis)of damage accumulation using S-N curves in linear form

(described with the Basquin function), models of the decreaseprocess of current endurance limit are considered. Gnilke (1987)

describes the change of the damage accumulation process using secondary S-N (Wöhler) lines as presented in figure 4. In this

model, the modification of the inflexion point position (N D(S),

σ D(S)) of the secondary S-N lines – whose theoretical bases are

developed by Gnilke (1980) - is taken into account gradually inproportion as the damage is accumulating. Schott (1975, 1990)

and Haibach (1989) present similar theories.

2.4. Conclusions for the modelling of damage accumulation

processSome partial conclusions that are useful for the modelling of

damage accumulation process are the next:1. The damage by material fatigue is a complex process

depending on numerous factors that are insufficiently known.

The modelling of the physical phenomenon at every moment is practically impossible, but there is the tendency of adapting the models so that it is taken into account.

2. The S-N diagrams represent the model of damage

accumulation process by uniform loading and must be used here as well as in the model by variable loading.

3. The decrease of the endurance limit pursuant to the pre-load to higher levels than this one has the consequence of small

stress levels contributing to the damaging process of thecycles.

3. Proposed damage accumulation model After taking into account the physical damage phenomenon, thedamage accumulation model should have the following

characteristics:a) the use of the failure probability as measurement of damage;

b) employing the S-N (Wöhler) diagram as a simple and precisemodel of the damage accumulation of the mechanical

components subjected to uniform cyclic loading;c) considering the pre-load influence by modifying the current

calculus stress.The cycle numbers equating principle by skipping from a stress

level to another is presented in figure 5. For this purpose the S-N curves modelled without inflexion point conforming to the ESOPE method (NF A 03-405:1991) are used.

There are numerous other known functions - excepting the Basquinone - that can be used to modelling S-N curves, see table 1. The

following symbols are used in this table: σ - stress; σ D – endurance

limit; N – cycle number; A, B, C, m0 – adjustable parameters.

These functions with more parameters offer a better approximationof the experimental points than the Basquin function. The authors

of this paper recommend the use of Stromayer and Weibull functions, because these are easily used and describe correctly thelower part of the S-N curve family, where the highest stress during

the gear operation there are.Conforming to Weibull (1961) and the French norm (NF A 03-

405:1991) the standard deviation of the endurance is constant

Figure 3. Relative endurance limit Dσ as function of relative

pre-stress cycle number, by Schott (1990), after Kommers

N - cycle number on Wöhler curve at the given stress; n - cycle

number made at pre-stress level σ i; σ D0 - initial endurance

limit; σ D=σ D/ σ D0 - current relative endurance limit.

1

0,8

0,6

0 0,5 1 n/N

)( 1,3σ

0σ

)( 1,2σ

0σ

)( 1,1σ

0σ

D0

D0

D0

+=

=

×=

ο

Dσ

a)

Secondary S-N (Wöhler) lines

S-N (Wöhler) line

N D(S) N D,0 (log)N [cycles]

(log)σ

σ D,0

σ D,min

σD(S)

b)(log) n [cycles]

σ D0 = x ⋅ σ a1

σ D(D) = xi ⋅ σ a1σ ai

σ D

σ

σ a1i = 1

i = 2

i = 3

i = 4

i = 5

i = 6

Initial damage

D = 0

Stressspectrum

with i = 6

blocks

S-N curve

Secondary

S-N curves

Figure 4. Models to modifying of S-N lines as consequence of damage accumulation

a) after Gnilke, 1987; b) Consequent Miner Rule (Haibach, 1989)

Figure 2. Endurance limit decrease as follow of

pre-stress σ 1 depassing the damage curve after French

(log) N [cycle]

σ D0

σ Dσ

σ 1

σ damage curve

(French line) S-N curve

nF



regardless of cycle number. Inoue (1994) has made the same

observation for the teeth root bending. Consequently, the family of S-N curves is easily modelled by translating the median curve by a

distance proportional to the standard deviation and the normal distribution quantile.

Table 1. Expressions used for S-N curves modelling (Weibull, 1961)

№ Author Expression

1 Basquin 1910] 0mN C −⋅=σ

2 Stromayer 1914] ( ) 0mD N C

−⋅=−σ σ

3 Weibull 1949 ( ) ( ) 0mD BN C −+⋅=−σ σ

4 Bastenaire 1957 ( )

D

mC 0DeAN

σ σ

σ σ

−⋅=

−⋅−

Besides the advantage of calculus simplicity, the described cycleequating method permits the bloc programs processing and of the

sequences with quasi-random mixed cycles, taking into account thestress cycles order and the differentiated damage accumulationmodelled with the S-N curves family.

The rupture probability (P r ) is used as a measure of the degree of

damage. Thus the predicted reliability distribution can be directlyobtained. The actual material damage can be characterized by thefailure producing probability (in this case rupture) but the damagehas (as physical process inherent stochastic) variations in a

statistical population.It is observed that the classical damage accumulation hypothesis

considered for the estimation of durability is associated withfailure probability (corresponding to the used S-N curve).The case of cycles having smaller tensions than the medium

endurance limit by rupture is solved in two ways:

a) by the previously described method in which the inferior part of S-N curve family is also used;

b) by simulation of step by step damaging capacity increasing of cycles inferior to median endurance limit.

As mentioned above, the decrease step by step of the endurancelimit was considered by a number of authors (Schott, 1975 and 1990; Gnilke, 1987; Haibach, 1989) by specific modifications of

the S-N curves (figure 4). On the other hand, the modelling without

inflexion of secondary S-N curves imposes the need to change the

shape of these ones in the domain above endurance limit.As a result of this analysis, a new method is proposed by the

authors - to simulate the damaging effect of increasing the number of cycles using the increase in the calculated stress instead of the

step-by-step decrease of the endurance limit. This is described by

the following expression that gives the corrected stress cσ

dependent upon the current stress σ :

B

A1

P %50,D

4

%50,D

r c

σ

σ

σ

σ σ ⋅

⋅+

+= ,

where P r is the failure probability, assimilated to the accumulated

damaging degree, σ D,50% represents the medium endurance limit and the coefficients A and B are material dependent. It is noted

that this proposed formula modifies the stress in the region of endurance limit of the S-N curves family.

4. ApplicationFor the purpose of testing the proposed calculus method, theresults obtained by this method were compared with:

• experimental results obtained with block program loading;• results of calculus by other methods.

The experiment was carried out on case carburised toothed wheel

(made up 18 MoCrNi 13 STAS 791-88) on a hydropulse testing machine using the loading block program given in table 2.

Table 2. The structure of loading block program

Block Size

1 2 3 4 5 6

Stress

[MPa]527 570 610 570 527 495

Cycle

number 1000 100 80 100 1000 2000

The obtained experimental points and the calculated durabilitycurves are represented in figure 6. The durability curves(corresponding to a failure probability of 50%) are calculated:a) on one hand - using the most used damage accumulation

models namely: Palmgren - Miner, Haibach, consequent Miner rule, Gnilke and Schott (Haibach, 1989; Gnilke, 1980;

Schott, 1975);b) and on the other hand - applying the proposed method.

The S-N curves used for determining the durability curves arerepresented in figure 7. It is noted that there is a significant difference between the two curve families especially in the

transition domain at the knee of the Basquin function (inlogarithmic coordinates). On the other hand this is the domain of

large stress levels that appear during the gear operation.

a) b)

Figure 5. Cycles equating principle by skip from a stress level to another:

a) temporal stress variation; b) cycle numbers equating and failure probability accumulation on S-N (Wöhler) diagram

N 2

1

2

3

4

5

σ

σ 4

σ 3

σ 2

σ 5

σ 1S-N

curves

t N 1 N 4 N' 1N' 3 N' 2 N 3 N 5 n

Figure 6. Durability curves determined by classical and proposed

method for the loading block program given in table 2

Haibach

Schott

Palmgren-Miner

Consequent

Miner Rule

Proposed

method

Gnilke

Experimental

points

σFlim [MPa]

950

900

850

800

750

700

650

600

550

500

450

2·10 5·104 105 5·105 106 N [ cycles]



It is interesting a new analysis using a block program adopted byother authors. The block program used by Haibach, (1989) and

Yang (1989) (table 3) has as particularity numerous (and large)loading blocks situated close to and below the endurance limit;

consequently, this block program differs in comparison with theprevious one indicated in the table 2.

The durability curves by different methods using the bloc programindicated in the table 3 are represented in the figure 8.

The following conclusions may be established by analysis of thesefigures:

a) in the domain of short durability all used calculus methodsproduce close results;

Table 3. The block program used for determining of durabilitycurves presented in figure 8

Block

Size 1 2 3 4 5 6 Relative

stress0.827 0.925 1 0.925 0.827 0.710

Cycle

number 680 70 9 70 680 5000

Block Step

7 8 9 10 11 12

Relative

stress0.592 0.475 0.358 0.475 0.592 0.710

Cyclenumber

23000 70000 302500

70000 23000 5000

b) the method we propose produces a calculated durabilitycurve with results closer to the experimental average due to

use of a more accurate S-N curves model;c) in the low domain of the S-N curves family the durability

lines are spaced out more;d) the durability values calculated by the classical methods are

displaced towards the smaller values. This can be explained

by the use of S-N curves modelled using the Basquin functiondescribing incorrectly the experimental dates in the kneearea (see Deckelmann, 1990);

e) the distribution calculated using the method we propose isasymptotic to endurance limit and the transition is without

discontinuities (very important in practical applications).This aspect is in agreement with the experimental resultspresented by Osterman (1971);

f) the curves determined by classical methods present great discontinuities caused by the calculated durability skip when

a cycle block is taken into account by exceeding theendurance limit. These distributions have a sudden skip toinfinity but below the endurance limit the durability is

infinite.

5. Conclusions1. A model of damage accumulation process based on the

physical aspects analysis of damage is proposed.2. The damage accumulation model uses the rupture probability

as damage measure, equates the cycle numbers (by skipping

from a stress level to another) using the S-N curves familymodelled without inflexion point and modifies the cycle

damaging effect dependent upon the accumulated damage.3. The model has been applied to two stress spectra (presented

as block programs) resulting the durability distributions.These ones have been compared with the distributionscalculated with classical theories and with experimental

results (for a stress level with the first spectrum). One would see a better approach of the experimental results, the absence

of the durability distributions’ discontinuity and their shape inconcordance with the experimental results from literature.

6. ReferencesDECKELMANN, G. Lebensdauervorbensage dynamisch

beanspruchter metallischer Bauteile unter Berücksichtigung desgekrümmten Verlaufs der Wohlerlinien, Fortschritt-Berichte VDI,

Reihe 5,Nr. 199, 1990.GNILKE, W. Lebensdauerberechnung der Maschinenelemente,VEB-Verlag Technik, Berlin, 1980.

GNILKE, W., KALLENBERG, J. 1987. Entwicklung empirischer Schadensgleichungen, Maschinenbautechnik 36, Nr.2, 1987, 72-74.

HAIBACH, E. Betriebsfestigkeit: Verfahren und Daten zur Bauteilberechnung, VDI-Verlag, Düsseldorf, 1989.

INOUE, K. a.o. Effects of surface condition on the bending strength of carburized gear teeth, Proc. of the 1994 International

Gearing Conf., University of Newcastle upon Tyne, U.K., 138-188.OSTERMAN, H. Verlauf der Lebensdauerlinie eines

Vergütungsstahls nach 8stufigen Programmversuch im Bereichoberhalb von 107 Lastspielen, Materialprüf. 13, 11, 1971, 389-391.SCHOTT, G. Vorschlag eines Verfahrens zur Berechnung der

Lebensdauer bei Mehrstufen - bzw. Kollektivbelastung,Problemseminar "Verkstoffermüdung", Dresden, 1975.

SCHOTT, G. Lebensdauerberechnung mit Werkstoffermüdungs-funktionen: Ermüdungsfestigkeit bei Mehrstufen-und

Randombeanspruchungen, Deutscher Verlag für Grundstoffindustrie, Leipzig, 1990.

WEIBULL, W. Fatigue Testing and Analysis of Results, Ed.Pergamon Press, Oxford, 1961.YANG, Q. Zuverlässigkeit von Zahnradgetrieben, Diss., Ruhr-

Universität, Bochum, 1989.DIN 3990:1987 Teil 3. Tragfähigkeitsberechnung von Stirnrädern.

Berechnung der Zahnfußtragfähigkeit, 1987.

Figure 7. S-N (Wöhler) curves used for durability calculus

modeled by Weibull function with ESOPE method;

modeled by Basquin function.

σFlim [MPa]

800

750

700

650

600

550

500

450

4002·104 10 10 10N [ cycles]

P r =10%

P r =50%

P r =90%

Figure 8. Durability curves determined by classical and proposed method for the loading block program given in table 3

σF [MPa]

Palmgren-Miner

Schott

Gnilke

Haibach

Consequent Miner Rule

Proposed Method

3·10510 10 108 109 N [cycles]

950

900

850

800

750

700

650

600

550

500

450



NF A03-405 : 1991. Produits métalliques. Essais de fatigue.

Traitement statistique des données, 1991.

on modeling the gear damage accumulation process

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