Mechanical Spring Reliability Assessments Based on FEA Generated Fatigue Stresses and Monte Carlo Simulated Stress/Strength Distributions

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1 Purdue Universit Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 4 Mechanical Spring Reliabilit Assessments Based on FEA Generated Fatigue Stresses and Monte Carlo Simulated Stress/Strength Distributions Jian Xiong Chen Copeland Corporation Wen Chen Yan Copeland Corporation Follow this and additional works at: Chen, Jian Xiong and Yan, Wen Chen, "Mechanical Spring Reliabilit Assessments Based on FEA Generated Fatigue Stresses and Monte Carlo Simulated Stress/Strength Distributions" (4). International Compressor Engineering Conference. Paper This document has been made available through Purdue e-pubs, a service of the Purdue Universit Libraries. Please contact epubs@purdue.edu for additional information. Complete proceedings ma be acquired in print and on CD-ROM directl from the Ra W. Herrick Laboratories at Herrick/Events/orderlit.html

2 C111, Page 1 MECHANICAL SPRING RELIABILITY ASSESSMENTS BASED ON FEA GENERATED FATIGUE STRESSES AND MONTE CARLO SIMULATED STRESS / STRENGTH DISTRIBUTIONS Jianxiong Chen, Ph.D., Sr. Engineering Specialist Wenzhen Yan, Sr. Lead Project Engineer Technical Services, Copeland Corporation, Emerson Climate Technologies 1675 West Campbell Road, P.O. Box 669, Sidne, Ohio , USA Tel.: 937/498-39, Fax: 937/ , jchen@copeland-corp.com ABSTRACT The reliabilit assessment of a mechanical spring is required to evaluate different spring designs for various spring deflection applications. For the fatigue failure mode, the spring reliabilit is predoantl dependent on the service fatigue stress and the fatigue strength of the spring. In realit, significant statistical scatters of both fatigue stress and fatigue strength are often inherent and inevitable. Therefore, the spring reliabilit assessment for highvolume production is a statistical interference analsis of fatigue stress and fatigue strength. In man engineering cases, neither the fatigue stress nor the statistical interference can be analticall solved; finite element analsis (FEA) and Monte Carlo simulations are thus emploed. Finite element analsis is used to generate the fatigue stresses as functions of spring deflections for displacement-controlled boundar conditions. The spring reliabilities as functions of fatigue life are predicted based on the Monte Carlo simulations of spring deflection and spring fatigue strength distributions. The methodolog described in this paper is equall good for force-controlled boundar conditions; and can be applied as a general probabilistic approach to evaluate the reliabilit of man other tpes of mechanical components. Application of this methodolog has the potential for reducing the product developmental cost and improving product robustness. 1. INTRODUCTION Finite element analsis (FEA) is a ver powerful and versatile technique to analze complicated structural problems. A finite element model is developed to calculate the fatigue stresses from the spring static preload to the spring dnamic deflections. For displacement-controlled boundar conditions, the fatigue stresses are explicitl expressed as functions of spring deflection using curve fitting of FEA results to quadratic polnomials. The Goodman relationship is used to include the mean stress effect on fatigue life in the analsis. Two input random variables, spring deflection and spring fatigue strength, are modeled b truncated normal distributions. The spring fatigue strength is characterized b the S-N approach. The spring fatigue stress is also a random variable and is calculated from the spring deflection using the quadratic polnomials of fatigue stress versus spring deflection. High-speed computers that quickl and economicall estimate the statistics of complex functions have led to the wide usage of Monte Carlo simulation methods in solving difficult engineering problems. In this analsis, since the fatigue stress distribution is not a standard probabilit densit function (i.e., Normal, Log normal, or Weibull distribution, Monte Carlo simulation becomes a suitable technique to solve this statistical interference problem. Monte Carlo simulations virtuall sample 1,, spring assemblies; and each has different deflections and fatigue strengths in a random manner according to their respective probabilit distributions. The output random variable of the Monte Carlo simulations is the spring fatigue life. Finall, the probabilit of failure (or the reliabilit) is calculated from Monte Carlo simulation results as functions of fatigue life. International Compressor Engineering Conference at Purdue, Jul 1-15, 4

3 C111, Page. FEA GENERATED FATIGUE STRESSES The FEA generated fatigue stresses are given in Figure 1 as functions of spring deflections for two different spring designs. For this displacement-controlled boundar condition problem, the FEA results can be curve fitted to ield explicit expressions of fatigue stress versus spring deflection: S ( x) = a 1 + a x + a 3 x (1) where a 1 through a 3 are coefficients; and x is the spring deflection. Quadratic polnomials are sufficientl accurate in representing the FEA results. Coefficients a 1 through a 3 are evaluated b a regression analsis. The stress at x = is due to spring static preload. The stress at x > is the dnamic fatigue stress under dnamic service displacements. Therefore, the imum stress is the stress at zero deflection (x = ), and the maximum stress max is the stress at the maximum spring deflection. Both imum and maximum stresses can be detered from Equation (1) as: = S( x = ) max = S( x = max) () The mean stress m and the stress amplitude a are calculated b following equations: max + m = max a = (3) The mean stress effect on fatigue life is included using Goodman relationship: a + m S e S u = 1 (4) where S u is the ultimate strength; and S e is the full reversed stress amplitude corresponding to the same life as that obtained with the stress condition a and m in Equation (3). The fatigue stress of a spring assembl is of constant amplitude; howeve the amplitude varies randoml among assemblies. Combining Equations (1) to (4), the full reversed stress amplitude can be finall detered b: [ S( x = max) S( x = ) ] S u [ S( x = max) + S( x = ) ] S e = (5) S u 3. STATISTICAL CHARACTERIZATIONS OF RANDOM VARIABLES The spring dnamic deflection is a linear tolerance stack-up of several components involved in the assembl. Each assembl has a distinct value of maximum spring deflection; different assemblies have different deflections. Therefore, the spring deflection is treated as a random variable. In general, the tolerances of these components involved in an assembl can be considered as random and statisticall independent to each other. It becomes reasonable to approximate the spring deflection distribution b a normal distribution according to the central limit theorem for sums. Furthermore, it is assumed that the manufacturing process is capable of maintaining the tolerance ranges to finite values for high volume production. Hence, the spring deflection distribution is truncated with a range of µ x ± 3 x. Note that µ x and x are mean and standard deviation of the normal distribution for spring International Compressor Engineering Conference at Purdue, Jul 1-15, 4

4 C111, Page 3 deflection respectivel. The Monte Carlo simulated histograms of spring deflection are shown in Figure for both large and small deflection ranges. The fatigue stress is calculated from the spring deflection using Equation (1). The fatigue stress is also a random variable; howeve its statistical characteristics are not explicitl expressed and are realized through Monte Carlo simulations. A truncated normal distribution is also assumed for the spring fatigue strength distribution with the range of µ s ± 3 s, where µ s and s are mean and standard deviation of the fatigue strength normal distribution. µ s is also the fatigue strength corresponding to 5% probabilit of failure and a full reversed loading situation (S e, p=5% ). In this analsis, the standard deviation s is set to 9% of the fatigue strength at 5 % probabilit of failure, or =. 9 µ (6) s s The Monte Carlo simulated histogram of spring fatigue strength is given in Figure 3. It is also assumed that the spring deflection and the fatigue strength are statisticall independent. The probabilit densit function of a truncated normal distribution is modified from a non-truncated normal distribution. The general mathematical expression of a truncated normal distribution of random variable has four distribution parameters (ANSYS Release 8. Documentation), namel a mean µ and a standard deviation of the non-truncated normal distribution, and the lower limit and upper limit max. The probabilit densit function of a truncated normal distribution is: f ( ) = for < or > max f ( ) = Φ max µ 1 Φ µ µ ϕ for max (7) The cumulative distribution function of the truncated normal distribution is: µ µ Φ Φ F ( ) = (8) µ µ Φ max Φ where φ(.) and Φ(.) are probabilit densit function and cumulative distribution function of the standard normal distribution, respectivel. φ(.) and Φ(.) are expressed as: ϕ( z) = Φ( z) = 1 z exp π z 1 z exp dz π for z + (9) The S-N curve in Figure 4 corresponds to 5% probabilit of failure and a full reversed loading situation. It is closel modeled within the available data range b a third-order exponential deca function as follows: International Compressor Engineering Conference at Purdue, Jul 1-15, 4

5 S N N N e, p= 5% = c + c 1 exp + c exp + exp c 3 t 1 t t (1) 3 C111, Page 4 where N is the fatigue life; and c to c 3 and t 1 to t 3 are coefficients detered b a regression analsis. The method presented in this paper for detering the fatigue stress can be similarl applied for force-controlled boundar conditions. 4. RELIABILITY ASSESSMENTS USING MONTE CARLO SIMULATIONS The limit state function is defined as: g( = N( N s (11) where r is a vector of the design parameters affecting the design strength, and s is a vector of the design parameters affecting the applied stress. In Equation (11), N s is the service fatigue life for which an estimate of reliabilit or probabilit of failure is required. N=N( is the random variable denoting fatigue life (the number of ccles to failure). Apparentl N is a function of design parameters. The limit state function is used to identif both the safe state and the failure state: Safe state : Failure state : g( = N( N s > g( = N( N s < (1) If P(.) denotes probabilit, the probabilit of failure can be calculated using the limit state function as: [ g( ) ] F ( N s ) = P s < (13) Alternativel, the probabilit of failure can be written as: F ( N s ) =... f ( x, x,..., x n ) dx dx... dx x n (14) where x i (i = 1,, n) are the generalized design parameters of interest. f x is the joint densit function of x i (i = 1,, n). The integration in Equation (14) is performed over the region of the failure state, and can not be evaluated analticall in this analsis. The direct Monte Carlo simulations technique is thus used to evaluate Equations (11) to (13). In Monte Carlo simulations, 1,, assemblies are virtuall sampled; all of them differ in a random manner from each other. Each assembl of a Monte Carlo simulation ccle has the spring deflection and the spring fatigue strength randoml generated according to their respective truncated normal distributions. The fatigue stress is then calculated from the spring deflection using Equation (1), and the full reversed fatigue stress amplitude is detered b Equation (5). Knowing spring fatigue stress and fatigue strength, the spring fatigue life is then evaluated. Because of the limited range of the S-N curve available, howeve in some cases the spring fatigue stresses fall outside the stress range of the S-N curve. The corresponding fatigue lives can not be exactl detered, but their upper or lower limit values are known: either above or below the life range of the S-N curve. Even though a complete spring fatigue life distribution analsis is usuall not feasible, this does not create an difficulties to estimate the probabilities of failure in the range of fatigue life defined b the S-N curve. Mathematicall, the probabilit of failure defined in Equations (11) and (13) can be estimated in Monte Carlo simulations as a function of N s b the following equation: International Compressor Engineering Conference at Purdue, Jul 1-15, 4

6 n ( N ) f s F( N s ) = (15) n ( N ) + n ( N ) f s s s Similarl, the reliabilit, as a function of N s, can be estimated b: n ( N ) R( N ) s s = s (16) n ( N ) + n ( N ) f s s s C111, Page 5 where n f (N s ) is the number of simulation ccles in which the spring fatigue lives are less than N s (g( <); and n s (N s ) is the number of simulation ccles in which the spring fatigue lives are greater than or equal to N s (g( ). The total number of simulation ccles is n f (N s ) + n s (N s ) = 1,,. It is apparent that F(N s ) + R(N s ) = 1 from Equations (15) and (16). The evaluations of F(N s ) and R(N s ) are carried out over the whole range of fatigue life defined b the S-N curve. The estimated probabilities of failure are expressed as functions of fatigue life in Figure 5. A sufficientl large number of Monte Carlo simulation ccles is needed in order to achieve converged results. Monte Carlo simulations with different total numbers of simulation ccles are tested, revealing that the total number of 1,, simulation ccles is enough to obtain converged results for this analsis. The accurac of converged results is basicall not a function of the total number of simulation ccles, but rather the uncertainties associated with the distributions of input random variables, including the uncertainties in the FEA modeling. The confidence levels are not specified for the statistics of input random variables, usuall impling the are approximatel average values. Accordingl, the confidence levels of the estimated probabilities of failure and the reliabilities are around 5%. 5. CONCLUSIONS The methodolog described in this paper has been successfull used to assess the reliabilities of different spring designs for various spring deflection range applications. A robust design has been identified and proposed at an earl stage of the product development. The combination of FEA and Monte Carlo simulations in this methodolog grants this methodolog to solve a wide range of complicated engineering problems. In fact, the methodolog can be used as a general probabilistic approach to analze man other kinds of mechanical components. This methodolog can also pla an important role in designing robust products while imizing the product developing cost. The accurac of the assigned distributions for the random variables can significantl affect the precision of the analsis results. Therefore, the integration of the experimental data into this methodolog is often necessar to more accuratel define the distributions. Bench fatigue tests can also be performed to calibrate the analsis results for at least one of the cases investigated. Howeve the end analsis results are alwas more meaningful when used as comparisons among different cases investigated. REFERENCES (1) ANSYS, INC., Theor Reference, ANSYS Release 8. Documentation. () Julie A. Bannantine, Jess J. Come & James L. Handrock, 199, Fundamentals of Metal Fatigue Analsis, Prentice Hall, Englewood Cliffs, New Jerse. (3) Sarath Jaatilleka, Geoffre Okogbaa, 3, Use of Accelerated Life Tests on Transmission Belts for Predicting Product Life, Identifing Better Design, Materials and Supplies, Proc. Ann. Reliabilit & Maintainabilit Smp., 3, Tampa, Florida, USA, p International Compressor Engineering Conference at Purdue, Jul 1-15, 4

7 C111, Page 6 (4) Charles Lipson & Narendra J. Sheth, 1973, Statistical Design and Analsis of Engineering Experiments, McGraw-Hill Book Compan. (5) Yongcang Zhang, Robert Rogers, & Ted Skrzszewski, 3, Reliabilit Prediction of Hdraulic Gasket Sealing, Proc. Ann. Reliabilit & Maintainabilit Smp., 3, Tampa, Florida, USA, p (6) Srikanth V. Thiruppukuzhi & Zenel Arslanoglu, 4, Designing Robust Products Using a Combination of Simulation, Experimental and Statistical Techniques, Proc. Ann. Reliabilit & Maintainabilit Smp., 4, Los Angles, California, USA, p (7) B. M. Patel, Vinod K. Nagpal, Shantaram S. Pai, & Lois J. Scaglione, 4, Probabilistic Structural Analsis of Ultra Efficient Engine Technolog Ceramic Matrix Composite Combustor Liners, Proc. Ann. Reliabilit & Maintainabilit Smp., 4, Los Angles, California, USA, p3-33. (8) John Mandel, 1984, The Statistical Analsis of Experimental Data, Dover Publications, Inc., New York. 75 Spring fatigue stress (ksi) Spring design #1 Spring design # Spring deflection (inch) Figure 1 FEA generated spring fatigue stresses as functions of spring deflections for two different spring designs (1 inch =.54 mete 1 ksi = 6,894,757 Pascal) International Compressor Engineering Conference at Purdue, Jul 1-15, 4

8 C111, Page Occurrence ,, M CS sam ples Spring deflection (inch) Occurrence ,, M CS sam ples Spring deflection (inch) Figure Monte Carlo simulated histograms of spring deflection for large spring deflection range (upper one) and small spring deflection range (lower one). 1 inch =.54 meter Occurrence ,, M CS sam ples Se/Se, p=5% Figure 3 Monte Carlo simulated histograms of spring fatigue strength International Compressor Engineering Conference at Purdue, Jul 1-15, 4

9 C111, Page 8 Alternating stress amplitude p=5% (ksi) Full reversed Number of ccles N Figure 4 Spring fatigue propert. The S-N curve represents the fatigue strength corresponding to 5% probabilit of failure and a full reversed loading situation (1 ksi = 6,894,757 Pascal).14.1 Spring design # -- small deflection range Spring design #1 -- small deflection range Probabilit of failure (%) Fatigue life (ccle) Probabilit of failure (%) Spring design # -- large deflection range Spring design #1 -- large deflection range Fatigue life (ccle) Figure 5 The estimated probabilities of spring failure from Monte Carlo simulations International Compressor Engineering Conference at Purdue, Jul 1-15, 4