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Research Papers

Reliability Methods for Bimodal Distribution With First-Order Approximation1

[+] Author and Article Information
Zhangli Hu

Department of Mechanical and
Aerospace Engineering,
Missouri University of Science and Technology,
258A Toomey Hall,
400 West 13th Street,
Rolla, MO 65409-0500
e-mail: zh3zd@mst.edu

Xiaoping Du

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

1Paper presented at the 2017 ASME International Design and Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE 2017) held on August 6-9, 2017, Cleveland, OH. Paper No. DETC2017- 67279.

Manuscript received November 5, 2017; final manuscript received April 15, 2018; published online August 14, 2018. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 5(1), 011005 (Aug 14, 2018) (8 pages) Paper No: RISK-17-1098; doi: 10.1115/1.4040000 History: Received November 05, 2017; Revised April 15, 2018

In traditional reliability problems, the distribution of a basic random variable is usually unimodal; in other words, the probability density of the basic random variable has only one peak. In real applications, some basic random variables may follow bimodal distributions with two peaks in their probability density. When binomial variables are involved, traditional reliability methods, such as the first-order second moment (FOSM) method and the first-order reliability method (FORM), will not be accurate. This study investigates the accuracy of using the saddlepoint approximation (SPA) for bimodal variables and then employs SPA-based reliability methods with first-order approximation to predict the reliability. A limit-state function is at first approximated with the first-order Taylor expansion so that it becomes a linear combination of the basic random variables, some of which are bimodally distributed. The SPA is then applied to estimate the reliability. Examples show that the SPA-based reliability methods are more accurate than FOSM and FORM.

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Figures

Grahic Jump Location
Fig. 1

A bimodal distribution with a mixture of two normal distributions

Grahic Jump Location
Fig. 2

Flowchart of the SPA methods

Grahic Jump Location
Fig. 3

A simple support beam

Grahic Jump Location
Fig. 4

PDF approximation using FOSM

Grahic Jump Location
Fig. 5

Contours of the limit-state function in the X-space

Grahic Jump Location
Fig. 6

Contours of the limit-state function in the U-space

Grahic Jump Location
Fig. 7

A speed reducer shaft

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