BK50A2200 Design Methodologies and Applications of Machine Element Design

1. BK50A2200 Design Methodologies and Applications of Machine Element Design Lecture 3 Reliability based machine design and lifetime analysis D.Sc Harri Eskelinen

2. 1 Introduction • Reliability based machine design and lifetime analysis gives the answer to the question: “What will be the expected lifetime of the component and what reliability level can be reached?” • The answer consists of two quantitative values: “Lifetime is 70 000 hours, probability of the failure is p=0.14 during that 70 000 hours period” • In traditional dimensioning of mechanical components only the value of required safety factor is calculated against the estimated loading: “Safety factor against the applying stress is n=2.5”

3. The main difference between traditional dimensioning and reliability based machine design and lifetime analysis can be simplified into following two points of view: • In traditional dimensioning the accepted over-dimensioning produces the desired “reliability”. • In reliability based machine design and lifetime analysis optimal dimensioning is based on the required “safe” lifetime and the probability of the failure during this lifetime is accepted on the estimated reliability level.

4. Completed reliability based machine design and lifetime analysis produces the following main results: • Lifetime and failure probability • Number of failures occurring per unit time =The failure rate function λ(t) • Possible failure modes in expected and sequential order • Possible consequences of the failure

5. 1 2 4 Collecting statistical data to describe the variations of the affecting stresses of the machine part. Collecting statistical data to describe the variations of the load bearing capacity of the machine Part (Resistance) Calculating probability distributions to describe the variations of the affecting stresses and the load bearing capacity 2 Main steps of reliability based design • The flow of reliability based design can be simplified into following main steps: 3 Tuning the size of the over-lapping area of the two probability distributions to establish the required reliability

6. To ensure 100% reliability resistance f(R) should be always higher than stress f(s) • In this case, even though the mean values µs and µr satisfy this condition µs < µr, the two distributions have yet an over-lapping area A. • Inside this area A the failure will happen

7. Reliability based design suffers from four difficulties: • How the variation of the affecting loads can be simplified to follow any known distribution function? • How the load bearing capacity (resistance) of different components of the construction can be simplified to follow any known distribution function? • How the affects of inaccuracy of different simplified dimensioning equations for typical machine parts can be taken into account? • How the so-called “human factor” can be included to the reliability analysis? • In many cases this means that the first step is to ensure “the reliability” of the reliability analysis it self!

8. Traditionally high values of safety factors have been used because of the following three aspects, which cause unexpected insecurity to the dimensioning of the component: • 1 Because of human mistakes the applying load can be much higher than expected during the dimensioning (e.g. loads of cranes, lifts, vehicles etc.) • 2 The total number of different types and values of load cycles can be difficult to estimate and the mean of stresses is not suitable to dimensioning criteria (e.g. in real life it is impossible to ensure that if a vehicle passes the bridge, the applying load to the bridge construction is equal in both cases…) • 3 Environmental conditions might change dramatically (e.g load due to heavy snow or jammed power transmission shaft)

9. In reliability based design these three aspects are handled in two ways: • By trying to estimate, what might be the probability, that there will be 50 kg extra load in a lift… • Or by establishing, what is the probability, that the lift, which is loaded to its upper limit, has a supporting cable with minimum accepted diameter and with minimum accepted tensile strength (lowest tolerance limits of these two values) • Reliability based design is able to take care of the latter case by utilizing the previously presented distribution based analysis but the former case can only be handled by substituting a safety factor, which is high enough.

10. The estimation of the stress (load) distributions can be made easier by: • Collecting data about the load peaks caused by some typical types of transmission systems or engines (e.g. electric motors or diesel-powered motors). In many cases the load peaks and their frequency can be presented in a form of the function depending of the speed) • Measuring affecting deformations of the component or vibrations inside the system to be able to compose load spectrum. This makes it possible to calculate the number, magnitude and frequency of each loading case and level.

11. Reasons for insecurity of dimensioning equations of typical machine parts (which leads usually to the use of high safety factors): • For combined load cases there are several dimensioning hypotheses available. The accuracy of each of them varies and is at its lowest near the ends of the application area. • e.g. pull + bend + shear of a shaft • fatigue analysis of welded structures • Typically dimensioning equations include assumptions, which do not fully correspond to the real world: • “Rigid” joint • Use of mass centres or concentrations

12. For reliability based design it is relatively easy to collect the following data: • 1. Variation of material properties • E.g. the following material properties of steels are allowed to vary inside the limits given in the standard’s data sheets: • Chemical composition (alloying) • Carbon content • Allowed amount of impurities • Porosity • Further on, variation of these items can cause variation in: • Strength values (tolerances given in standards) • The results of heat treatments (e.g. hardness) • Suitability for welding

13. 2. Statistical results from heat treatments: • E.g. following data can be collected: • Variation of the oven temperature • Variation of the inner atmosphere of the oven • Variation of the heating/cooling velocities (temperatures) at the different parts of the component • E.g. from a hardening process the following practical design data for reliability analysis can be collected: • Variation of surface hardness of a component (e.g. gears) • Variation of the thickness of the hardened surface (e.g. against wear) • Magnitude of residual stresses

14. 3. Statistical data from manufacturing accuracy: • E.g. collected data from turning process: • Variation of shaft diameter per each manufacturing set • Variation of surface roughness per each manufacturing set • This data affects directly on: • Variation of load bearing cross-section area of the shaft • Magnitude of stress peaks on the shaft • Power transmission capacity of shaft-hub-joints (based on fits) • Values of fixed stress values due to surface roughness (needed in fatigue diagrams) • Fatigue analysis (each surface defect can behave as an initial crack

15. 2.1 Principles to combine statistical data of stress f(S) and resistance variation f(R) • Let us, at first, make a brief “qualitative” analysis for a power transmission shaft described below (by applying to the previous three aspects): • Shaft with different diameters for roller bearings, seals and a hub-shaft-joint • A gear is milled directly on a shaft • Hub-shaft joint for torque transmission is based on the use of an appropriate fit • Material is QT-steel

16. The load bearing capacity (resistance) is at its lowest, when: • 1. Variation of different shaft diameters • The diameter at the critical cross section is at its minimum • Stress components are at their maximum • Difference between the larger and smaller diameter is at its maximum • Stress concentration and stress peak get their maximum values • The diameter which is in key-role against deformation (either bending or torsion) is at its minimum • Shaft stiffness is at its minimum, which could lead to contact errors between the gears and different wear phenomena on gears and roller bearings.

17. 2. Carbon content is at its minimum and the amount of alloying components, which improve hardenability is at its lowest (but each of them is inside the standard variation): • Compression stress due to successful heat treatment is at its lowest • This reduces the fatigue strength of the shaft • Hardness and the thickness of the hardened surface are at their lowest • Wear resistance is at its lowest • Ductility of the gear is at its maximum, but this is difficult to utilize in practical dimensioning, because gear dimensioning is based on the use of mean values of hardness and ductility.

18. 3. Meaning of the surface roughness variation: • Surface roughness is given by Ra-value, which is the standard deviation of the profile. This allows relatively large occasional faults on the surface. • When Ra is at its maximum: • Fatigue strength decreases • When Ra is at its minimum: • Shaft seals can get stuck on the shaft and be ripped off due to elevated temperature and lack of lubrication • When occasional /single faults are at their maximum: • Faults can behave like initial cracks of fatigue failure • Individual faults can cause accelerated wear of shaft seals

19. Case-hardened Hardeness 58 +/- 3 HRC

20. 4. Power transmission capacity is at its minimum, when • The compression of the fit is at its lowest: • Shaft diameter is at its lowest (hub diameter is at its largest) • Clearance is at its maximum due to smoothing of the surface roughnesses • 5. Stress at the hub is at its maximum, when • Clearance is at its minimum • More aspects could be derived from gear and gear contact… • Most important is to realize there is a contradiction between some of the aspects: • In most of cases reliability design can be regarded as optimum design, in which balance is searched between product’s properties and requirements.

21. 2.2 Calculations with distributions • Let us refer back to the picture of stress f(S) and resistance f(R) (= load bearing capacity) distribution… • If there is no over-lapping area, no failure could ever take place • Let us assume that these distributions over-lap. To be able to calculate the failure probability we should calculate the remainder between stress f(S) and resistance f(R).

22. Would it happen, that both of these distributions can be expresses with the so-called normal distribution function, all required calculations became relatively simple: • It is possible to calculate how failure probability depends on the mean values and standard deviations of stress and resistance • Also the new calculated distribution will fit well to the normal distribution function and easy-to-use thumb-rules based on tabulated data can be applied

23. Let us use the following symbols: • µr = mean value of resistance (distribution) • µs = mean value of stress (distribution) • σr = standard deviation of resistance • σs = standard deviation of stress • zp = co-efficient for normal distribution function (depending on required probability level) • p = required probability level (0…1) • Without going deeper into the mathematics, we get the relationship for practical calculations:

24. p zp • Example: The required reliability is 90% (probability for failure is 10%). It is known that standard deviation of yield strength is σr = 0.03×µr and standard deviation of the stress is σs = 0.12×µsHow much higher should materials yield strength be than the expected applying stress? • At first we read the design values from normal distribution function data table: • We get from the table: when p=0.9 then zp= 1.282

25. The result means that material’s yield strength should be 16% higher than the mean value of the applying stress to achieve the required reliability 90% • Results illustrates the required “over-dimensioning” to ensure the reliability. • By substituting K= µr/ µs we get: • (K-1)2=1.282 × (0.032 ×K2+0.122) • By solving this we get two roots: 1.16 (and 0.84)

26. “Safety Factor” 1.16 with the risk of failure 4 % “Safety Factor” 1.16 (Mean value) Load Yield strength 203 Mpa 235 MPa ± 4% ± 2% Load Yield strength 203 Mpa 235 Mpa

27. 2.2.1 Weibull distribution • In most of cases the load bearing capacity of the machine part, which remains near the lowest section of the normal distribution is quite exceptional, but the resistance, which is near the highest values of the distribution is desired. • Quality-oriented manufacturing and material science makes it possible a) to improve the mean values and reduce the standard deviations. • In critical components or constructions the required reliability level is increased to avoid e.g. enormous costs or health or environmental risks due to possible failure. • In these cases mathematically fitted Weibull distribution describes the reliability (or failure probability) of the component much better than normal distribution. • Weibull distribution is typically used e.g. roller bearing lifetime analysis.

28. Unwanted values ”High-quality” values Improved Mean value

29. standard deviation (σ) • mean (µ)

30. Weibull distribution Normal distribution

31. Depending of the values of the three parameters η, β and γ Weibull distribution can be fitted to follow quite different types of data sets.

32. Quite close to normal distribution Roller bearings

33. Case example of how to utilize Weibull distribution • According to reliability tests it has been possible to derive a simple estimation equation of the life time and reliability of rolling bearings based on the use of Weibull probability distribution function. The simplified form is as follows: • Where • R = required reliability (0…1) • L = ideal (or theoretic) life time • L10 = required life time related to R (required “over-dimensioning”)

34. Let us assume that • R = required reliability = 0.99 • L = ideal (or theoretic) life time = 20 000 h • L10 = required life time related to R should be estimated (required “over-dimensioning”), • The previous equation gives first for the reliability R • … and then for required life time L10

35. 3 Reliability engineering and modelling • One of the most essential stages of reliability based dimensioning is to construct a model, which describes the connections and impacts of different reliability aspects: • This model can describe either the whole system or the construction of the specified machine (e.g. reasons for malfunctions of the whole packaging line or reasons for the malfunctions of an individual packaging machine) • This model can describe also the connections between different reliability aspects of a single machine part (e.g. different reasons (with corresponding probabilities) for gear wear) • This model can be a combination on these two previously mentioned aspects (a failure mode table of a gear to analyze the reasons and consequences if the torque transmission capacity is decreased or prevent)

36. 3.1 Some tools for reliability engineering • A) Traditional statistics • Mean values and standard deviations • Correlations between different variables/ aspects • Linear regression and other mathematical connections • Comparisons with reference values • The use of too simple tools (like mean values combined with ”expected” normal distributions) might lose important information. In many cases this kind of information gives only some guiding results for reliability based dimensioning.

37. B) Applied probability theory • Histograms and density distributions and their curve fittings • Derived probability distributions and their curve fittings • E.g. Weibull distributions • Calculations with distributions • Principal questions: • What is the combined probability of the affecting aspects, which could lead to the failure? • What is the most likely failure mode? • What are the most important reasons for possible failure?

38. C) Methods of risk analysis • Anticipation of potential damages • Reaction matrix • Malfunctioning analysis • Analyzing dangerous scenarios • Analyzing consequences • Failure and impact analysis • Cause and effect chains • Event tree analysis • Fault tree analysis • Taguchi methods

39. Anticipation of potential damages • Most critical components or system parts are collected based on experience and the goal is try to ”forecast” the possible problems, which might take place based on these results • Suffers from subjective assumptions • Reaction matrix • The aim is to collect factor pairs (on/off- analysis), which together can cause failures. • Can not recognize the combinations of several affecting factors • It is possible to recognize only the direct affects, which could lead to the failure • Makes it easier to find those factor pairs, between which correlation calculations are useful

40. Malfunctioning analysis • Finds the reasons for malfunctioning of the system, which depend on the failure of individual machine parts • Analyzing dangerous scenarios • Finds those situations, in which the machine part must be changed or the system must be repaired due to e.g. wear, too large clearance, lack of lubrication etc., though the final breakdown has not happened yet but could be expected to happen soon.

41. Analyzing consequences • Typically a list or a table of already recognized reasons for failures of different machine parts with analyzed consequences. • Usually can not be used directly for reliability or lifetime analysis, but it supports the use of other tools (e.g. drawing fault trees) • Failure and impact analysis • A large construction is divided into smaller parts and all the impacts affecting on the failure are analyzed part by part. • This analysis shows the impacts only from the bottom up.

42. Cause and effect chains • Different malfunctioning levels of the system are presented in a form of sequential ”events” (=”chain”). • Only the critical cause-effect-connections are presented without any probability values. • E.g. engine + clutch + gear + transmission shaft • Event tree analysis • Otherwise like the previous cause and effect chains but chains are now formed according to events, which are sequentially time-depended • E.g. affects of wear at different levels of the system during the estimated lifetime

43. Fault tree analysis • Also the probabilities of different failures at different system levels are added into cause and effect chains • It is possible to calculate probabilities of different fault combinations • Sometimes it is difficult to exactly express when the final breakdown has happened (e.g. wear phenomena) • Taguchi methods • Each factor, which has no influence on the failure, will be isolated from the analysis factor by factor. Finally only the real reasons will remain as the result of the analysis.

44. 3.1.1 Fault trees • In addition to the use of distribution calculations fault trees are among the most used tools for reliability engineering • Symbols used in fault trees are standardized • The chart is formed from bottom up. The TOP-event at the highest shows the final breakdown of the system/ machine/ construction. • The chart illustrates the whole system. This makes it possible to recognize all the possible paths, which might lead to the failure. • The “weak links” of the system can be recognized (the parts, which could breakdown for several individual reasons)

45. By utilizing the chart it is possible to design and construct safe and spare functions for important functions of the system. • The individual events should be specified in details so that the probabilities of each event can be established for further analysis. • The final failure should be named exactly, so that the reason can be found. Standardized failure mode analysis can be applied.(Deformation, Creep, Yielding, Fatigue, Fracture, Corrosion, Wear, Melting, Buckling, etc.) • The basic two questions are: • Is it just impossible to use the component (e.g. it suffers from too large deformations or disturbing vibrations during the use though the breakdown is far away…) OR • Has the component already broken down?

46. Examples of failure mode analysis

47. Detailed failure mode analysis

48. Camshaft failure mechanism in general: • Abrasive wear, scuffing, plastic deformation and pitting • Screw joint failure mechanism in general: • Fatigue crack, corrosion

50. Symbols used in fault trees