- Email: [email protected]
Practical reliability-based design of gear pairs
Mechanism and Machine Theory 38 (2003) 1363–1370 www.elsevier.com/locate/mechmt
Practical reliability-based design of gear pairs Y.M. Zhang a
a,*
, Qiaoling Liu a, Bangchun Wen
b
Department of Mechanics, Nanling Campus, Jilin University, Changchun 130025, PR China School of Mechanical Engineering, Northeastern University, Shenyang 110006, PR China
b
Received 28 October 2002; received in revised form 16 May 2003; accepted 18 May 2003
Abstract Techniques from perturbation method and reliability-based design theory are employed to present the practical and effective method for the reliability-based design of gear pairs. The theoretical formulae of reliability-based design of gear pairs are obtained. The corresponding program can be used to obtain the reliability-based design parameters of gear pairs accurately and quickly. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction Reliability and its significance in engineering design is a rapidly growing field. The safety and reliability of an engineering component is invariably the principal technical objective of an engineering design. As these must invariably be done in the presence of uncertainty, the proper reliability or safety may only be stated in the probability. Indeed, consistent levels of safety and reliability may be achieved only if the reliability-based design is based on such the probabilistic methods of reliability. In the recent years, much research has been done to quantify uncertainties in engineering systems and their combined effect on the reliability (for example [1–7]). The interfix reliability-based design of mechanical components has been described (for example [8,9]). Theoretically, these uncertainties are modeled as random variables governed by joint probability density or distribution functions. In practice, the exact joint probability density functions are often unavailable or difficult to obtain for reasons of insufficient data. Not infrequently, the available data may only be sufficient to evaluate the first few moments such as the mean, variance and correlations.
*
Corresponding author. E-mail address: [email protected] (Y.M. Zhang).
0094-114X/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0094-114X(03)00092-2
1364
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
Gear drives are widely adapted to many kinds of mechanical equipment. The complex construction of a gear pairs introduces many factors which influence gear pairs reliability. Varying work condition, various loads and process all contribute to the difficulty in resolving the problem. By now, there is no definite conclusion on reliability design of gear pairs. It is importance to research the reliability-based design of gear pairs. Although the reliability problem of gear pairs can be solved by reliability experiments, it is hard to carry out the experiment under the limits of the experiment conditions, manpower, resource and time. Therefore, a numerical method should be introduced. During the last two decades, there has been considerable activity in the area of engineering research. The reliability-based design of a gear drive has been described. For a thorough summary of the research in the reliability-based design of gear pairs, see the papers [10– 17]. This paper focuses on extension of the probabilistic perturbation method for reliability-based design of gear pairs. Using the perturbation method and the reliability-based design theory, this paper proposes a numerical method to calculate and design the reliability of gear pairs. The statistical techniques and methods for these purposes are widely illustrated in this paper. This methodology can be used to obtain the reliability-based design information of cylindrical gear pairs accurately and quickly. According to the results, the method is convenient and practical. This sophisticated formulation is easily amenable to computational procedures.
2. The perturbation method of the reliability-based design Most engineering systems are designed without the benefit of complete information; consequently, the assurance of performance can seldom be perfect. Moreover, many decisions during the design process are invariably made under conditions of uncertainty. Therefore, there is invariably some chance of nonperformance or failure and associated adverse consequences; hence, risk is often unavoidable. Traditionally, the reliability of engineering systems is achieved through the use of coefficients of safety and by adopting conservative assumptions in the design process. The traditional approach is difficult to quantify and lacks the logical basis for addressing uncertainties; consequently, the level of safety or reliability cannot be assessed quantitatively. Moreover, for new systems in which there is no prior basis for calibration, the problem of assuring performance would obviously be difficult. This following numerical method employs the perturbation method and reliability-based design theory to systematically develop the formulae for calculating and designing of the reliability of gear pairs. A fundamental problem in reliability analysis is the computation of the multi-fold integral of the reliability R Z fX ðX Þ dX ð1Þ R¼ gðXÞ>0
in which fX ðXÞ denotes the probability density function of the vector of random parameters T X ¼ ðX1 ; X2 ; . . . ; Xn Þ and gðXÞ defines the state function, representing the safe state and failure state
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
gðXÞ 6 0 gðXÞ > 0
failure state safe state
1365
ð2Þ
where gðXÞ ¼ 0 is the limit-state equation, representing an n-dimensional surface that may be called the ‘‘limit-state surface’’ or the ‘‘failure surface’’. The vector of random parameters X and the state function gðXÞ are expanded as X ¼ X d þ eX p
ð3Þ
gðXÞ ¼ gd ðXÞ þ egp ðXÞ
ð4Þ
where e is a small parameter. Subscript d in Eqs. (3) and (4) denotes certain part of the random parameters, while subscript p denotes the random part having a zero mean value in the random parameters. Obviously, it is necessary for the value of the random part to be smaller than the value of the certain part. Both sides of Eqs. (3) and (4) are evaluated about the mean value of random variables as follows EðXÞ ¼ EðX d Þ þ eEðX p Þ ¼ X d E½gðXÞ ¼ E½gd ðXÞ þ eE gp ðXÞ ¼ gd ðXÞ
ð5Þ ð6Þ
Similarly, according to the Kronecker algebra [18,19], both sides of Eqs. (3) and (4) are evaluated about the variance of the random variables as follows n o h i VarðXÞ ¼ E ½X EðXÞ½2 ¼ e2 X ½2 ð7Þ p n o n o ½2 Var½gðXÞ ¼ E ½gðXÞ EðgðXÞÞ½2 ¼ e2 E gp ðXÞ ð8Þ Using the Kronecker product, the Kronecker power ðÞ½2 is represented as 2 3 a11 A a12 A a1q A 6 a21 A a22 A a2q A 7 6 7 A½2 ¼ A A ¼ 6 . .. .. .. 7 4 .. . . . 5 ap1 A a11 A . . .
ð9Þ
apq A
where the Kronecker power of Aðp qÞ is denoted by A A and is a p2 q2 matrix, and represents the Kronecker product. By expanding the state function gp ðXÞ to first-order approximation in a Taylor series of vectorvalued functions and matrix-valued functions at a point EðXÞ ¼ X d , which is on the failure surface gp ðX d Þ ¼ 0, that is ogd ðXÞ Xp oX T Substituting Eq. (10) into Eq. (8) yields " # ½2 ½2 og ðXÞ ogd ðXÞ d ½2 2 Xp ¼ VarðXÞ Var½gðXÞ ¼ e E oX T oX T gp ðXÞ ¼
ð10Þ
ð11Þ
where VarðXÞ is the variance matrix which includes all of the variance and covariance of the random parameters.
1366
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
The reliability index [20] is defined as b¼
lg E½gðXÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rg Var½gðXÞ
ð12Þ
It may be emphasized that the first-order approximation of lg and rg , derived above, must be evaluated at the mean values ðlX1 ; lX2 ; . . . ; lXn Þ. In some approximate sense, the reliability index may be directly used as a measure of reliability. If the distributions of the original random variables are normal, the distance from the ÔminimumÕ tangent plane to the failure surface may be used to approximate the actual failure surface, and corresponding reliability may be represented as follows R ¼ UðbÞ
ð13Þ
where Uð Þ is the standard normal distribution function. The available first few moments (such as the mean, variance and correlations) of the original random variable vector X can be sufficient to calculate the reliability. If the distribution of the original random variable vector X is non-normal, the corresponding reliability, using Eq. (13), may be more conservative to system reliability analysis.
3. The reliability-based design with the flank contact strength Calculated contact stress of gear pairs is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ft u 1 KA KV KH b KH a rH ¼ ZH ZE Ze Zb bd1 u
ð14Þ
where rated tangential tooth force at transverse pitch Ft b active face width work condition coefficient KA dynamic load coefficient KV longitudinal load distribution coefficient KH b transverse load distribution coefficient KH a nodal field coefficient ZH elastic coefficient ZE contact ratio coefficient Ze helix angle coefficient Zb pinion pitch diameter d1 u gear ratio Flank contact fatigue strength of tooth faces is defined as r0H lim ¼ rH lim ZN ZR ZV ZL ZW ZX where rH lim experimental flank contact fatigue strength life coefficient ZN
ð15Þ
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
1367
tooth fineness coefficient velocity coefficient lubricant coefficient work harden coefficient size coefficient The above 19 fundamental parameters should be regarded as random variables except the gear ratio u. According to stress–strength interference theory, the state function is defined as ð16Þ gðXÞ ¼ r0H lim rH
ZR ZV ZL ZW ZX
where
T X ¼ rH lim ZN ZR ZV ZL ZW ZX ZH ZE Ze Zb Ft bd1 KA KV KH b KH a 18 1
ð17Þ
the mean values, variance and covariances of the original random parameter are known. Differentiation of the state function gðXÞ with respect to the original random parameter vector X gives ogðXÞ og og og og og og og og og og og og og og og og og og ¼ orH lim oZN oZR oZV oZL oZW oZX oZH oZE oZe oZb oFt ob od1 oKA oKV oKH b oKH a 1 18 oX T ð18Þ Substituting the equations and known conditions above into expressions of a mean matrix and a variance matrix, the mean value and variance of the state function are studied out. Then, substituting the mean value and variance into expressions of the reliability index b and reliability R, the reliability index b and reliability R can be determined. Furthermore, if the reliability R of the gear pairs is given, the reliability index b is consulted from the table of standard normal probability. Apparently, the algebra equation for designing d1 which is derived from lg ¼ brg is the nonlinear transcendental equation. The solution of the nonlinear transcendental equation may be obtained through the common iterative algorithm. There is a problem about choosing initial values. The gear designers should have some experiences about the initial values, and these initial values can be used to iterate. Then, the perfect results can be obtained. 4. The reliability-based design with the dedendum bending strength Calculated bending stress of gear pairs is defined as rF ¼ YFa YSa Ye Yb where mn YFa YSa Ye Yb
Ft KA KV KF b KF a bmn
normal module tooth form factor dedendum stress concentration coefficient contact ratio factor helix angle coefficient
the other parameters are the same as the forward.
ð19Þ
1368
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
Dedendum bending fatigue strength is defined as r0F lim ¼ rF lim YST YNT Yd relT YR rel T YX where rF lim YST YNT Yd relT YR rel T YX
ð20Þ
experimental gear bending fatigue strength experimental gear dedendum stress concentration coefficient life coefficient relative sensitive coefficient relative surface condition coefficient size coefficient
The above 17 original parameters are treated as random variables. According to stress–strength interference theory, the state function is defined as gðXÞ ¼ r0F lim rF
ð21Þ
where T X ¼ rF lim YST YNT Yd relT YR rel T YX YFa YSa Ye Yb Ft bmn KA KV KF b KF a 17 1
ð22Þ
the mean values, variance and covariances of the original random parameter are known. Differentiation of the state function gðXÞ with respect to the original random parameter vector X gives ogðXÞ og og og og og og og og og og og og og og og og og ¼ orF lim oYST oYNT oYdrelT oYRrelT oYX oYFa oYSa oYe oYb oFt ob omn oKA oKV oKF b oKF a 1 17 oX T ð23Þ In the same way, we can calculate the reliability of gear pairs and carry out the reliability design about mn . 5. Statistical characteristics of original random parameters 1. Every mean value of random variables is obtained by using the methods proposed in the standard regulations (such as looking up in the interfix handbook). 2. If the experimental data are not available, the variance and covariance of the random parameters can be estimated on the following principles: (a) The standard deviations of mechanical property parameters are determined by the variation coefficient C, in general, C ¼ 0:05; (b) The standard deviations of geometry parameters are determined using the tolerance standard and the 3r principle. In general, the tolerance size is 0.015 times the nominal size, if the tolerance size is three times the standard deviation, the variation coefficient C ¼ 0:005; (c) The standard deviations of loads are determined by the experiment; (d) The standard deviations of material strength are determined using the experiment or the material handbook; (e) The standard deviations of other values are determined using their error (±0.1–0.15) and three times the standard deviations, and then the variation coefficients are C ¼ 0:033;
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
1369
(f) The correlation of the parameters is determined by correlation coefficient q. If no experimental data are available, it can be determined approximately by experience according to the practical conditions and mathematical statistics theory. In general, the larger the correlation coefficient is, the stronger the linear correlation among the parameters is. 6. Numerical example Using the method in accordance with the standard regulations or looking up in interfix handbook, get the mean values of each parameter of gear pairs in a machine, on the above principles, get the standard deviation of each parameter too. The mean values and standard deviations are denoted as follows: Ft ¼ ð34644; 519:66Þ N, KA ¼ ð1; 0:033Þ, KV ¼ ð1:484; 0:1613Þ, KH b ¼ ð1:68; 0:05544Þ, 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mm, d1 ¼ ð148:75; 0:74375Þ KF b ¼ ð1:603; 0:052899Þ, KH a ¼ KF a ¼ ð1:16; 0:03828Þ, 2b ¼ ð200;p mm, mn ¼ ð4; 0:02Þ mm, ZH ¼ ð2:32; 0:0116Þ, ZE ¼ ð189:8; 9:49Þ N=mm2 , Ze ¼ ð0:81; 0:00405Þ, Zb ¼ ð0:957; 0:004785Þ, rH lim ¼ ð1300; 156Þ N/mm2 , ZN ¼ ð1; 0:033Þ, ZR ¼ ð1:03; 0:03399Þ, ZV ¼ ð1:04; 0:03432Þ, ZL ¼ ð0:92; 0:03036Þ, ZW ¼ ð1; 0:033Þ, ZX ¼ ð1; 0:033Þ, YF a1 ¼ ð2:36; 0:07788Þ, YF a2 ¼ ð2:14; 0:07062Þ, YS a1 ¼ ð1:75; 0:05775Þ, YS a2 ¼ ð1:94; 0:06402Þ, Ye ¼ ð0:715; 0:003575Þ, Yb ¼ ð0:8; 0:004Þ, YST ¼ ð2:1; 0:0693Þ, rF lim ¼ ð310; 62Þ N/mm2 , YNT ¼ ð1; 0:033Þ, Yd rel T1 ¼ ð0:99; 0:03267Þ, Yd rel T2 ¼ ð1:01; 0:03333Þ, YR rel T1 ¼ YR rel T2 ¼ ð1:065; 0:035145Þ, YX1 ¼ YX2 ¼ ð1; 0:033Þ. (1) Design based on the flank contact stress strength According to the known data, we can get b ¼ 2:9685
R ¼ 0:9985
If the given value of R is 0.99, the reliability index can be b ¼ 2:33 and sequentially the mean and standard deviation of the pinion pitch circle minimum diameter are calculated d1 ¼ ð109:5639; 0:5478196Þ mm (2) Design based on the dedendum bending stress strength According to the known data, the reliability index and reliability of pinion and gear pairs can be determined respectively b1 ¼ 2:754 R1 ¼ 0:9963 b2 ¼ 2:705 R1 ¼ 0:9966 If the given value of R is 0.99, the mean value and standard deviation of minimum normal module are calculated based on the pinion design mn ¼ ð3:4191; 0:017095Þ mm:
7. Conclusion This paper probes into the method of the reliability of cylindrical gear pairs. Using the method, the reliability of gear pairs can be obtained quantitatively. The reliability-based design of gear pairs can be provided on the basis of reliability-based design, so it is an important exploration of the reliability research of gear pairs. The results of the design and calculation accord well with the
1370
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
practical operation of gear pairs. Therefore, this method provides an effective and reliable approach to assessing the reliability of gear pairs.
Acknowledgements We would like to express our appreciation to the Chinese National Natural Science Foundation (number: 50175043, 19990510), to the 973 Project Foundation of China (number: 1998020320) and to the Jilin Natural Science Foundation for supporting this research for supporting this research.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
E. Vanmarcke, M. Shinozuka, S. Nakagiri, G. Schueller, M. Grigoriu, Structural Safety 3 (1986) 143–166. R.A. Ibrahim, Applied Mechanics Review 15 (3) (1987) 309–328. H. Benaroya, M. Rebak, Applied Mechanics Review 41 (1988) 201–213. Y.M. Zhang, S.H. Chen, Q.L. Liu, T.Q. Liu, Computers & Structures 59 (3) (1996) 425–429. Y.M. Zhang, B.C. Wen, S.H. Chen, Mathematics and Mechanics of Solids 1 (4) (1996) 445–461. B.C. Wen, Y.M. Zhang, Q.L. Liu, International Journal of Nonlinear Dynamics 15 (2) (1998) 179–190. Y.M. Zhang, B.C. Wen, Q.L. Liu, Computer Methods in Applied Mechanics and Engineering 165 (4) (1998) 223– 231. Y.M. Zhang, Q.L. Liu, Proceedings of the Institution of Mechanical Engineers Part D, Journal of Automobile Engineering 216 (D6) (2002) 455–471. Y.M. Zhang, Q.L. Liu, Proceedings of the Institution of Mechanical Engineers Part C, Journal of Mechanical Engineering Science 216 (C2) (2002) 179–182. S.S. Rao, Journal of Mechanical Design, Transactions of the ASME 101 (4) (1979) 625–632. S.S. Rao, G. Das, Journal of Mechanisms, Transmissions, and Automation in Design 106 (1) (1984) 17–22. E.M. Al-Shareedah, H. Alawi, Mechanism & Machine Theory 22 (1) (1987) 13–20. T.S. Pan, S.S. Rao, Journal of Mechanisms, Transmissions, and Automation in Design 110 (3) (1988) 348–353. S.S. Rao, M. Tjandra, Reliability Engineering & System Safety 46 (2) (1994) 159–169. U. Nikhage, Izvestiya Akademii Nauk. Energetika, Nov–Dec (6) (1994) 44–54. Q.J. Yang, International Journal of Fatigue 18 (3) (1996) 171–177. P.A. Mayo, Gear Technology 15 (5) (1998) 8. W.J. Vetter, SIAM Review 15 (1973) 352–369. J.W. Brewer, IEEE Transactions on Circuits and Systems CAS-25 (1978) 772–781. A.H.S. Ang, W.H. Tang, Probability Concepts in Engineering Planning and Design, in: Decision, Risk, and Reliability, vol. 2, John Wiley & Sons, Inc., New York, 1975.
Practical reliability-based design of gear pairs Y.M. Zhang a
a,*
, Qiaoling Liu a, Bangchun Wen
b
Department of Mechanics, Nanling Campus, Jilin University, Changchun 130025, PR China School of Mechanical Engineering, Northeastern University, Shenyang 110006, PR China
b
Received 28 October 2002; received in revised form 16 May 2003; accepted 18 May 2003
Abstract Techniques from perturbation method and reliability-based design theory are employed to present the practical and effective method for the reliability-based design of gear pairs. The theoretical formulae of reliability-based design of gear pairs are obtained. The corresponding program can be used to obtain the reliability-based design parameters of gear pairs accurately and quickly. Ó 2003 Elsevier Ltd. All rights reserved.
1. Introduction Reliability and its significance in engineering design is a rapidly growing field. The safety and reliability of an engineering component is invariably the principal technical objective of an engineering design. As these must invariably be done in the presence of uncertainty, the proper reliability or safety may only be stated in the probability. Indeed, consistent levels of safety and reliability may be achieved only if the reliability-based design is based on such the probabilistic methods of reliability. In the recent years, much research has been done to quantify uncertainties in engineering systems and their combined effect on the reliability (for example [1–7]). The interfix reliability-based design of mechanical components has been described (for example [8,9]). Theoretically, these uncertainties are modeled as random variables governed by joint probability density or distribution functions. In practice, the exact joint probability density functions are often unavailable or difficult to obtain for reasons of insufficient data. Not infrequently, the available data may only be sufficient to evaluate the first few moments such as the mean, variance and correlations.
*
Corresponding author. E-mail address: [email protected] (Y.M. Zhang).
0094-114X/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0094-114X(03)00092-2
1364
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
Gear drives are widely adapted to many kinds of mechanical equipment. The complex construction of a gear pairs introduces many factors which influence gear pairs reliability. Varying work condition, various loads and process all contribute to the difficulty in resolving the problem. By now, there is no definite conclusion on reliability design of gear pairs. It is importance to research the reliability-based design of gear pairs. Although the reliability problem of gear pairs can be solved by reliability experiments, it is hard to carry out the experiment under the limits of the experiment conditions, manpower, resource and time. Therefore, a numerical method should be introduced. During the last two decades, there has been considerable activity in the area of engineering research. The reliability-based design of a gear drive has been described. For a thorough summary of the research in the reliability-based design of gear pairs, see the papers [10– 17]. This paper focuses on extension of the probabilistic perturbation method for reliability-based design of gear pairs. Using the perturbation method and the reliability-based design theory, this paper proposes a numerical method to calculate and design the reliability of gear pairs. The statistical techniques and methods for these purposes are widely illustrated in this paper. This methodology can be used to obtain the reliability-based design information of cylindrical gear pairs accurately and quickly. According to the results, the method is convenient and practical. This sophisticated formulation is easily amenable to computational procedures.
2. The perturbation method of the reliability-based design Most engineering systems are designed without the benefit of complete information; consequently, the assurance of performance can seldom be perfect. Moreover, many decisions during the design process are invariably made under conditions of uncertainty. Therefore, there is invariably some chance of nonperformance or failure and associated adverse consequences; hence, risk is often unavoidable. Traditionally, the reliability of engineering systems is achieved through the use of coefficients of safety and by adopting conservative assumptions in the design process. The traditional approach is difficult to quantify and lacks the logical basis for addressing uncertainties; consequently, the level of safety or reliability cannot be assessed quantitatively. Moreover, for new systems in which there is no prior basis for calibration, the problem of assuring performance would obviously be difficult. This following numerical method employs the perturbation method and reliability-based design theory to systematically develop the formulae for calculating and designing of the reliability of gear pairs. A fundamental problem in reliability analysis is the computation of the multi-fold integral of the reliability R Z fX ðX Þ dX ð1Þ R¼ gðXÞ>0
in which fX ðXÞ denotes the probability density function of the vector of random parameters T X ¼ ðX1 ; X2 ; . . . ; Xn Þ and gðXÞ defines the state function, representing the safe state and failure state
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
gðXÞ 6 0 gðXÞ > 0
failure state safe state
1365
ð2Þ
where gðXÞ ¼ 0 is the limit-state equation, representing an n-dimensional surface that may be called the ‘‘limit-state surface’’ or the ‘‘failure surface’’. The vector of random parameters X and the state function gðXÞ are expanded as X ¼ X d þ eX p
ð3Þ
gðXÞ ¼ gd ðXÞ þ egp ðXÞ
ð4Þ
where e is a small parameter. Subscript d in Eqs. (3) and (4) denotes certain part of the random parameters, while subscript p denotes the random part having a zero mean value in the random parameters. Obviously, it is necessary for the value of the random part to be smaller than the value of the certain part. Both sides of Eqs. (3) and (4) are evaluated about the mean value of random variables as follows EðXÞ ¼ EðX d Þ þ eEðX p Þ ¼ X d E½gðXÞ ¼ E½gd ðXÞ þ eE gp ðXÞ ¼ gd ðXÞ
ð5Þ ð6Þ
Similarly, according to the Kronecker algebra [18,19], both sides of Eqs. (3) and (4) are evaluated about the variance of the random variables as follows n o h i VarðXÞ ¼ E ½X EðXÞ½2 ¼ e2 X ½2 ð7Þ p n o n o ½2 Var½gðXÞ ¼ E ½gðXÞ EðgðXÞÞ½2 ¼ e2 E gp ðXÞ ð8Þ Using the Kronecker product, the Kronecker power ðÞ½2 is represented as 2 3 a11 A a12 A a1q A 6 a21 A a22 A a2q A 7 6 7 A½2 ¼ A A ¼ 6 . .. .. .. 7 4 .. . . . 5 ap1 A a11 A . . .
ð9Þ
apq A
where the Kronecker power of Aðp qÞ is denoted by A A and is a p2 q2 matrix, and represents the Kronecker product. By expanding the state function gp ðXÞ to first-order approximation in a Taylor series of vectorvalued functions and matrix-valued functions at a point EðXÞ ¼ X d , which is on the failure surface gp ðX d Þ ¼ 0, that is ogd ðXÞ Xp oX T Substituting Eq. (10) into Eq. (8) yields " # ½2 ½2 og ðXÞ ogd ðXÞ d ½2 2 Xp ¼ VarðXÞ Var½gðXÞ ¼ e E oX T oX T gp ðXÞ ¼
ð10Þ
ð11Þ
where VarðXÞ is the variance matrix which includes all of the variance and covariance of the random parameters.
1366
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
The reliability index [20] is defined as b¼
lg E½gðXÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rg Var½gðXÞ
ð12Þ
It may be emphasized that the first-order approximation of lg and rg , derived above, must be evaluated at the mean values ðlX1 ; lX2 ; . . . ; lXn Þ. In some approximate sense, the reliability index may be directly used as a measure of reliability. If the distributions of the original random variables are normal, the distance from the ÔminimumÕ tangent plane to the failure surface may be used to approximate the actual failure surface, and corresponding reliability may be represented as follows R ¼ UðbÞ
ð13Þ
where Uð Þ is the standard normal distribution function. The available first few moments (such as the mean, variance and correlations) of the original random variable vector X can be sufficient to calculate the reliability. If the distribution of the original random variable vector X is non-normal, the corresponding reliability, using Eq. (13), may be more conservative to system reliability analysis.
3. The reliability-based design with the flank contact strength Calculated contact stress of gear pairs is defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ft u 1 KA KV KH b KH a rH ¼ ZH ZE Ze Zb bd1 u
ð14Þ
where rated tangential tooth force at transverse pitch Ft b active face width work condition coefficient KA dynamic load coefficient KV longitudinal load distribution coefficient KH b transverse load distribution coefficient KH a nodal field coefficient ZH elastic coefficient ZE contact ratio coefficient Ze helix angle coefficient Zb pinion pitch diameter d1 u gear ratio Flank contact fatigue strength of tooth faces is defined as r0H lim ¼ rH lim ZN ZR ZV ZL ZW ZX where rH lim experimental flank contact fatigue strength life coefficient ZN
ð15Þ
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
1367
tooth fineness coefficient velocity coefficient lubricant coefficient work harden coefficient size coefficient The above 19 fundamental parameters should be regarded as random variables except the gear ratio u. According to stress–strength interference theory, the state function is defined as ð16Þ gðXÞ ¼ r0H lim rH
ZR ZV ZL ZW ZX
where
T X ¼ rH lim ZN ZR ZV ZL ZW ZX ZH ZE Ze Zb Ft bd1 KA KV KH b KH a 18 1
ð17Þ
the mean values, variance and covariances of the original random parameter are known. Differentiation of the state function gðXÞ with respect to the original random parameter vector X gives ogðXÞ og og og og og og og og og og og og og og og og og og ¼ orH lim oZN oZR oZV oZL oZW oZX oZH oZE oZe oZb oFt ob od1 oKA oKV oKH b oKH a 1 18 oX T ð18Þ Substituting the equations and known conditions above into expressions of a mean matrix and a variance matrix, the mean value and variance of the state function are studied out. Then, substituting the mean value and variance into expressions of the reliability index b and reliability R, the reliability index b and reliability R can be determined. Furthermore, if the reliability R of the gear pairs is given, the reliability index b is consulted from the table of standard normal probability. Apparently, the algebra equation for designing d1 which is derived from lg ¼ brg is the nonlinear transcendental equation. The solution of the nonlinear transcendental equation may be obtained through the common iterative algorithm. There is a problem about choosing initial values. The gear designers should have some experiences about the initial values, and these initial values can be used to iterate. Then, the perfect results can be obtained. 4. The reliability-based design with the dedendum bending strength Calculated bending stress of gear pairs is defined as rF ¼ YFa YSa Ye Yb where mn YFa YSa Ye Yb
Ft KA KV KF b KF a bmn
normal module tooth form factor dedendum stress concentration coefficient contact ratio factor helix angle coefficient
the other parameters are the same as the forward.
ð19Þ
1368
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
Dedendum bending fatigue strength is defined as r0F lim ¼ rF lim YST YNT Yd relT YR rel T YX where rF lim YST YNT Yd relT YR rel T YX
ð20Þ
experimental gear bending fatigue strength experimental gear dedendum stress concentration coefficient life coefficient relative sensitive coefficient relative surface condition coefficient size coefficient
The above 17 original parameters are treated as random variables. According to stress–strength interference theory, the state function is defined as gðXÞ ¼ r0F lim rF
ð21Þ
where T X ¼ rF lim YST YNT Yd relT YR rel T YX YFa YSa Ye Yb Ft bmn KA KV KF b KF a 17 1
ð22Þ
the mean values, variance and covariances of the original random parameter are known. Differentiation of the state function gðXÞ with respect to the original random parameter vector X gives ogðXÞ og og og og og og og og og og og og og og og og og ¼ orF lim oYST oYNT oYdrelT oYRrelT oYX oYFa oYSa oYe oYb oFt ob omn oKA oKV oKF b oKF a 1 17 oX T ð23Þ In the same way, we can calculate the reliability of gear pairs and carry out the reliability design about mn . 5. Statistical characteristics of original random parameters 1. Every mean value of random variables is obtained by using the methods proposed in the standard regulations (such as looking up in the interfix handbook). 2. If the experimental data are not available, the variance and covariance of the random parameters can be estimated on the following principles: (a) The standard deviations of mechanical property parameters are determined by the variation coefficient C, in general, C ¼ 0:05; (b) The standard deviations of geometry parameters are determined using the tolerance standard and the 3r principle. In general, the tolerance size is 0.015 times the nominal size, if the tolerance size is three times the standard deviation, the variation coefficient C ¼ 0:005; (c) The standard deviations of loads are determined by the experiment; (d) The standard deviations of material strength are determined using the experiment or the material handbook; (e) The standard deviations of other values are determined using their error (±0.1–0.15) and three times the standard deviations, and then the variation coefficients are C ¼ 0:033;
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
1369
(f) The correlation of the parameters is determined by correlation coefficient q. If no experimental data are available, it can be determined approximately by experience according to the practical conditions and mathematical statistics theory. In general, the larger the correlation coefficient is, the stronger the linear correlation among the parameters is. 6. Numerical example Using the method in accordance with the standard regulations or looking up in interfix handbook, get the mean values of each parameter of gear pairs in a machine, on the above principles, get the standard deviation of each parameter too. The mean values and standard deviations are denoted as follows: Ft ¼ ð34644; 519:66Þ N, KA ¼ ð1; 0:033Þ, KV ¼ ð1:484; 0:1613Þ, KH b ¼ ð1:68; 0:05544Þ, 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mm, d1 ¼ ð148:75; 0:74375Þ KF b ¼ ð1:603; 0:052899Þ, KH a ¼ KF a ¼ ð1:16; 0:03828Þ, 2b ¼ ð200;p mm, mn ¼ ð4; 0:02Þ mm, ZH ¼ ð2:32; 0:0116Þ, ZE ¼ ð189:8; 9:49Þ N=mm2 , Ze ¼ ð0:81; 0:00405Þ, Zb ¼ ð0:957; 0:004785Þ, rH lim ¼ ð1300; 156Þ N/mm2 , ZN ¼ ð1; 0:033Þ, ZR ¼ ð1:03; 0:03399Þ, ZV ¼ ð1:04; 0:03432Þ, ZL ¼ ð0:92; 0:03036Þ, ZW ¼ ð1; 0:033Þ, ZX ¼ ð1; 0:033Þ, YF a1 ¼ ð2:36; 0:07788Þ, YF a2 ¼ ð2:14; 0:07062Þ, YS a1 ¼ ð1:75; 0:05775Þ, YS a2 ¼ ð1:94; 0:06402Þ, Ye ¼ ð0:715; 0:003575Þ, Yb ¼ ð0:8; 0:004Þ, YST ¼ ð2:1; 0:0693Þ, rF lim ¼ ð310; 62Þ N/mm2 , YNT ¼ ð1; 0:033Þ, Yd rel T1 ¼ ð0:99; 0:03267Þ, Yd rel T2 ¼ ð1:01; 0:03333Þ, YR rel T1 ¼ YR rel T2 ¼ ð1:065; 0:035145Þ, YX1 ¼ YX2 ¼ ð1; 0:033Þ. (1) Design based on the flank contact stress strength According to the known data, we can get b ¼ 2:9685
R ¼ 0:9985
If the given value of R is 0.99, the reliability index can be b ¼ 2:33 and sequentially the mean and standard deviation of the pinion pitch circle minimum diameter are calculated d1 ¼ ð109:5639; 0:5478196Þ mm (2) Design based on the dedendum bending stress strength According to the known data, the reliability index and reliability of pinion and gear pairs can be determined respectively b1 ¼ 2:754 R1 ¼ 0:9963 b2 ¼ 2:705 R1 ¼ 0:9966 If the given value of R is 0.99, the mean value and standard deviation of minimum normal module are calculated based on the pinion design mn ¼ ð3:4191; 0:017095Þ mm:
7. Conclusion This paper probes into the method of the reliability of cylindrical gear pairs. Using the method, the reliability of gear pairs can be obtained quantitatively. The reliability-based design of gear pairs can be provided on the basis of reliability-based design, so it is an important exploration of the reliability research of gear pairs. The results of the design and calculation accord well with the
1370
Y.M. Zhang et al. / Mechanism and Machine Theory 38 (2003) 1363–1370
practical operation of gear pairs. Therefore, this method provides an effective and reliable approach to assessing the reliability of gear pairs.
Acknowledgements We would like to express our appreciation to the Chinese National Natural Science Foundation (number: 50175043, 19990510), to the 973 Project Foundation of China (number: 1998020320) and to the Jilin Natural Science Foundation for supporting this research for supporting this research.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
E. Vanmarcke, M. Shinozuka, S. Nakagiri, G. Schueller, M. Grigoriu, Structural Safety 3 (1986) 143–166. R.A. Ibrahim, Applied Mechanics Review 15 (3) (1987) 309–328. H. Benaroya, M. Rebak, Applied Mechanics Review 41 (1988) 201–213. Y.M. Zhang, S.H. Chen, Q.L. Liu, T.Q. Liu, Computers & Structures 59 (3) (1996) 425–429. Y.M. Zhang, B.C. Wen, S.H. Chen, Mathematics and Mechanics of Solids 1 (4) (1996) 445–461. B.C. Wen, Y.M. Zhang, Q.L. Liu, International Journal of Nonlinear Dynamics 15 (2) (1998) 179–190. Y.M. Zhang, B.C. Wen, Q.L. Liu, Computer Methods in Applied Mechanics and Engineering 165 (4) (1998) 223– 231. Y.M. Zhang, Q.L. Liu, Proceedings of the Institution of Mechanical Engineers Part D, Journal of Automobile Engineering 216 (D6) (2002) 455–471. Y.M. Zhang, Q.L. Liu, Proceedings of the Institution of Mechanical Engineers Part C, Journal of Mechanical Engineering Science 216 (C2) (2002) 179–182. S.S. Rao, Journal of Mechanical Design, Transactions of the ASME 101 (4) (1979) 625–632. S.S. Rao, G. Das, Journal of Mechanisms, Transmissions, and Automation in Design 106 (1) (1984) 17–22. E.M. Al-Shareedah, H. Alawi, Mechanism & Machine Theory 22 (1) (1987) 13–20. T.S. Pan, S.S. Rao, Journal of Mechanisms, Transmissions, and Automation in Design 110 (3) (1988) 348–353. S.S. Rao, M. Tjandra, Reliability Engineering & System Safety 46 (2) (1994) 159–169. U. Nikhage, Izvestiya Akademii Nauk. Energetika, Nov–Dec (6) (1994) 44–54. Q.J. Yang, International Journal of Fatigue 18 (3) (1996) 171–177. P.A. Mayo, Gear Technology 15 (5) (1998) 8. W.J. Vetter, SIAM Review 15 (1973) 352–369. J.W. Brewer, IEEE Transactions on Circuits and Systems CAS-25 (1978) 772–781. A.H.S. Ang, W.H. Tang, Probability Concepts in Engineering Planning and Design, in: Decision, Risk, and Reliability, vol. 2, John Wiley & Sons, Inc., New York, 1975.