0
Research Papers

Time-Dependent Mechanism Reliability Analysis With Envelope Functions and First-Order Approximation

[+] Author and Article Information
Xiaoping Du

Department of Mechanical
and Aerospace Engineering,
Missouri University of Science and Technology,
290D Toomey Hall,
400 West 13th Street,
Rolla, MO 65409-0500
e-mail: dux@mst.edu

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received November 3, 2013; final manuscript received May 4, 2014; published online June 2, 2014. Assoc. Editor: David Gorsich.

J. Mech. Des 136(8), 081010 (Jun 02, 2014) (7 pages) Paper No: MD-13-1504; doi: 10.1115/1.4027636 History: Received November 03, 2013; Revised May 04, 2014

This work develops an envelope approach to time-dependent mechanism reliability defined in a period of time where a certain motion output is required. Since the envelope function of the motion error is not explicitly related to time, the time-dependent problem can be converted into a time-independent problem. The envelope function is approximated by piecewise hyperplanes. To find the expansion points for the hyperplanes, the approach linearizes the motion error at the means of random dimension variables, and this approximation is accurate because the tolerances of the dimension variables are small. The expansion points are found with the maximum probability density at the failure threshold. The time-dependent mechanism reliability is then estimated by a multivariable normal distribution at the expansion points. As an example, analytical equations are derived for a four-bar function generating mechanism. The numerical example shows the significant accuracy improvement.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Issac, K. K., 1993, “A Nondifferentiable Optimization Algorithm for Constrained Minimax Linkage Function Generation,” ASME J. Mech. Des., 115(4), pp. 978–988. [CrossRef]
Aviles, J. V., Hernández, A., and Amezua, E., 1995, “Nonlinear Optimization of Planar Linkages for Kinematic Syntheses,” Mech. Mach. Theory, 30(4), pp. 501–518. [CrossRef]
Simionescu, P. A., and Beale, D., 2002, “Optimum Synthesis of the Four-Bar Function Generator in Its Symmetric Embodiment: The Ackermann Steering Linkage,” Mech. Mach. Theory, 37(12), pp. 1487–1504. [CrossRef]
Mariappan, J., and Krishnamurty, S., 1996, “A Generalized Exact Gradient Method for Mechanism Synthesis,” Mech. Mach. Theory, 31(4), pp. 413–421. [CrossRef]
Rao, A. C., 1979, “Synthesis of 4-Bar Function-Generators Using Geometric Programming,” Mech. Mach. Theory, 14(2), pp. 141–149. [CrossRef]
Mallik, A. K., Ghosh, A., and Dittrich, G., 1994, Kinematic Analysis and Synthesis of Mechanisms, CRC-Press, Boca Raton, FL.
Zhu, J., and Ting, K.-L., 2000, “Uncertainty Analysis of Planar and Spatial Robots With Joint Clearances,” Mech. Mach. Theory, 35(9), pp. 1239–1256. [CrossRef]
Baumgarten, J. R., and Werff, K. V. D., 1985, “A Probabilistic Study Relating to Tolerancing and Path Generation Error,” Mech. Mach. Theory, 20(1), pp. 71–76. [CrossRef]
S. G.Dhande, J. C., 1973, “Analysis and Synthesis of Mechanical Error in Linkages—A Stochastic Approach,” ASME J. Eng. Ind., 95(3), pp. 672–676. [CrossRef]
Wei-Liang, X., and Qi-Xian, Z., 1989, “Probabilistic Analysis and Monte Carlo Simulation of the Kinematic Error in a Spatial Linkage,” Mech. Mach. Theory, 24(1), pp. 19–27. [CrossRef]
Dubowsky, S., Norris, M., Aloni, M., and Tamir, A., 1984, “An Analytical and Experimental Study of the Prediction of Impacts in Planar Mechanical Systems With Clearances,” ASME J. Mech., Des., 106(4), pp. 444–451. [CrossRef]
Parenti-Castelli, V., and Venanzi, S., 2005, “Clearance Influence Analysis on Mechanisms,” Mechanism and Machine Theory, 40(12), pp. 1316–1329. [CrossRef]
Tsaia, M.-J., and Lai, T.-H., 2008, “Accuracy Analysis of a Multi-Loop Linkage With Joint Clearances,” Mech. Mach. Theory, 43(9), pp. 1141–1157. [CrossRef]
Zhen, H., 1987, “Error Analysis of Position and Orientation in Robot Manipulators,” Mech. Mach. Theory, 22(6), pp. 577–581. [CrossRef]
Rajagopalan, S., and Cutkosky, M., 2003, “Error Analysis for the in-Situ Fabrication of Mechanisms,” ASME J. Mech. Des., 125(4), pp. 809–822. [CrossRef]
Sergeyev, V. I., 1974, “Methods for Mechanism Reliability Calculation,” Mech. Mach. Theory, 9(1), pp. 97–106. [CrossRef]
Bhatti, P., 1989, “Probabilistic Modeling and Optimal Design of Robotic Manipulators,” Ph.D. thesis, Purdue University, West Lafayette, IN.
Shi, Z., Yang, X., Yang, W., and Cheng, Q., 2005, “Robust Synthesis of Path Generating Linkages,” Mech. Mach. Theory, 40(1), pp. 45–54. [CrossRef]
Shi, Z., 1997, “Synthesis of Mechanical Error in Spatial Linkages Based on Reliability Concept,” Mech. Mach. Theory, 32(2), pp. 255–259. [CrossRef]
Du, X., Venigella, P. K., and Liu, D., 2009, “Robust Mechanism Synthesis With Random and Interval Variables,” Mech. Mach. Theory, 44(7), pp. 1321–1337. [CrossRef]
Du, X., 1996, “Reliability Synthesis for Mechanism,” Mach. Des., 13(1), pp. 8–11.
Rao, S. S., and Bhatti, P. K., 2001, “Probabilistic Approach to Manipulator Kinematics and Dynamics,” Reliab. Eng. Syst. Saf., 72(1), pp. 47–58. [CrossRef]
Liu, T. S., and Wang, J. D., 1994, “A Reliability Approach to Evaluating Robot Accuracy Performance,” Mech. Mach. Theory, 29(1), pp. 83–94. [CrossRef]
Bhatti, P. K., and Rao, S. S., 1988, “Reliability Analysis of Robot Manipulators,” ASME J. Mech. Des., 110(2), pp. 175–181. [CrossRef]
Kim, J., Song, W.-J., and Kang, B.-S., 2010, “Stochastic Approach to Kinematic Reliability of Open-Loop Mechanism With Dimensional Tolerance,” Appl. Math. Model., 24(5), pp. 1225–1237. [CrossRef]
Bowlin, A. P., Renaud, J. E., Newkirk, J. T., and Patel, N. M., 2007, “Reliability-Based Design Optimization of Robotic System Dynamic Performance,” ASME J. Mech. Des., 129(4), pp. 449–455. [CrossRef]
Yaofei, T., Jianjun, C., Chijiang, Z., and Yongqin, C., 2007, “Reliability Analysis of Kinematic Accuracy for the Elastic Slider-Crank Mechanism,” Front. Mech. Eng. China, 2(2), pp. 214–217. [CrossRef]
Howell, L. L., Rao, S. S., and Midha, A., 1994, “Reliability-Based Optimal Design of a Bistable Compliant Mechanism,” ASME J. Mech. Des., 116(4), pp. 1115–1221. [CrossRef]
Zhang, J., Wang, J., and Du, X., 2011, “Time-Dependent Probabilistic Synthesis for Function Generator Mechanisms,” Mech. Mach. Theory, 46(9), pp. 1236–1250. [CrossRef]
Sudret, B., 2008, “Analytical Derivation of the Outcrossing Rate in Time-Variant Reliability Problems,” Struct. Infrastruct. Eng., 4(5), pp. 353–362. [CrossRef]
Li, J., Chen, J.-B., and Fan, W.-L., 2007, “The Equivalent Extreme-Value Event and Evaluation of the Structural System Reliability,” Struct. Saf., 29(2), pp. 112–131. [CrossRef]
Chen, J.-B., and Li, J., 2007, “The Extreme Value Distribution and Dynamic Reliability Analysis of Nonlinear Structures With Uncertain Parameters,” Struct. Saf., 29(2), pp. 77–93. [CrossRef]
Lutes, L. D. A. S., 2004, Random Vibrations: Analysis of Structural and Mechanical Systems, Elsevier, Butterworth, Heinemann, Burlington, MA.
Li, J., and Mourelatos, Z. P., 2009, “Time-Dependent Reliability Estimation for Dynamic Problems Using a Niching Genetic Algorithm,” ASME J. Mech. Des., 131(7), pp. 1009–1022. [CrossRef]
Wang, Z., and Wang, P., 2012, “A Nested Extreme Response Surface Approach for Time-Dependent Reliability-Based Design Optimization,” ASME J. Mech. Des., 134(12), p. 121007. [CrossRef]
Breitung, K., 1988, “Asymptotic Crossing Rates for Stationary Gaussian Vector Processes,” Stoch. Process. Appl., 29(2), pp. 195–207. [CrossRef]
Breitung, K., 1993, “Asymptotic Approximations for the Crossing Rates of Poisson Square Waves,” Proceedings of the Conference on Extreme Value Theory and Applications, NIST Special Publication, Gaithersburg, MD, Vol. 3, pp. 75–80.
Rackwitz, R., 1997, “Time-Variant Reliability for Non-Stationary Processes by the Outcrossing Approach,” Probabilistic Methods for Structural Design, Solid Mechanics and Its Applications, Vol. 56, Springer, The Netherlands, pp. 245–260.
Schrupp, K., and Rackwitz, R., 1988, “Outcrossing Rates of Marked Poisson Cluster Processes in Structural Reliability,” Appl. Math. Model., 12(5), pp. 482–490. [CrossRef]
Andrieu-Renaud, C., Sudret, B., and Lemaire, M., 2004, “The PHI2 Method: A Way to Compute Time-Variant Reliability,” Reliab. Eng. Syst. Saf., 84(1), pp. 75–86. [CrossRef]
Lutes, L. D., and Sarkani, S., 2009, “Reliability Analysis of Systems Subject to First-Passage Failure,” NASA Langley Research Center, Technical Report No. NASA/CR-2009-215782.
Hagen, O., and Tvedt, L., 1991, “Vector Process Out-Crossing as Parallel System Sensitivity Measure,” J. Eng. Mech., 117(10), pp. 2201–2220. [CrossRef]
Rice, S. O., 1944, “Mathematical Analysis of Random Noise,” Bell Syst. Tech. J., 23(3), pp. 282–332. [CrossRef]
Rackwitz, R., 2001, “Reliability Analysis—A Review and Some Perspectives,” Struct. Saf., 23(4), pp. 365–395. [CrossRef]
Hu, Z., and Du, X., 2013, “Time-Dependent Reliability Analysis With Joint Upcrossing Rates,” Struct. Multidiscip. Optim., 48(5), pp. 893–907. [CrossRef]
Wang, Z., Mourelatos, Z. P., Li, J., Baseski, I., and Singh, A., 2014, “Time-Dependent Reliability of Dynamic Systems Using Subset Simulation With Splitting Over a Series of Correlated Time Intervals,” ASME J. Mech. Des., 136(6), p. 061008. [CrossRef]

Figures

Grahic Jump Location
Fig. 3

Motion error at the means of dimension variables

Grahic Jump Location
Fig. 2

Four-bar function generator mechanism

Grahic Jump Location
Fig. 4

Probability if failure on [95.5 deg,215.5 deg]

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In