A sample of 100 specimens of a titanium alloy was subjected to a strain-life fat

Question

A sample of 100 specimens of a titanium alloy was subjected to a strain-life fatigue test to determine time to crack initiation. The test was run up to a limit of 100,000 cycles. The observed times of crack initiation (in units of 1000 of cycles) were 18, 32, 39, 53, 59, 68, 77, 78, and 93. No crack had initiated in any of the other 91 specimens. a. Compute a nonparametric estimate of the pdf and reliability function. b. Plot the data on Weibull paper. Use the plot to obtain an estimate of the Weibull shape parameter beta and alpha, and comment on the adequacy of the weibull distribution. The reliability of a turbine blade can be represented by the following expression: R(t) = (1 + r/t_0)^2, 0 lessthanorequalto t lessthanorequalto t_0. where t_0 is the maximum life of the blade. a. Show that the blades are experiencing wearout. b. Compute MTTF for a maximum life of 3000 h. The following censored data reflect failure times in months, of a new computer scanner. Determine the reliability of this scanner over its 2-year warranty period. 9, 35+, 17, 28, 25+, 37+, 28+, 27. Use Rank Increment rank statistic to plot a Weibull pdf using the following multiple censored data. Which part of the bathtub curve are these data most likely coming from? The failure time (in hour) of a component is lognormally distributed with parameters mu_1 = 4 and sigma_r = 09 a. What is the MTTF for this component? b. When should the component be replaced, if the minimum required reliability for the component is 0.95? c. What is the value of the hazard function at this time (i.e., for the time calculated in part (b))? d. What is the mean residual life at this time if the component has already survived to this time (i.e., for the time calculated in part (b))? On surface of a highly stressed motor shaft there are five crack initiation points observed after the shaft is tested at 6000 rpm for 5h. a. Calculate the point estimate of a crack initiation per cycle (per turn of the shaft). b. Determine the 90% two-sided confidence interval of the probability of a crack initiation per cycle. c. What is the point estimate of the mean-cycles-to-crack initiation? d. What is the probability of no crack initiation in a test of 2000rpm for 10h? A household appliance is advertised as having more than a 10-year life. If the following is its time-to-failure pdf, determine the reliability of a new appliance at 10 year. f(r) = 0.1(1+0.05r)^3 t greaterthanorequalto 0. What is its MTTF when it is new, and what is its MTTF when a 1-year warranty period ends, assuming it still works?

Explanation / Answer

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Solution of 3.56:

R(t) = P(T> t | T>10) = (1 - F(t)) / (1-F(1) )

F(t) = integralt0 f(t).dt = integralt0 0.1 / (1 +0.05t)3.dt = integralt0 800 / (20 + t)3.dt

= 1 - 400 / (t+20)2

For next 10 years is equal to t = 11

F(11) = 1 - 400 / (11+20)2 =1 - 400 / 312

F(1) =1 - 400 / 441 = 41 / 441

R(t=11) = (1 - F(11)) / (1-F(1) ) = ( 400 / 312 )/ ( 400 / 441 )

=441 / 961 = 0.46

Without warranty period R(t) = 1- F(t) =1 - 1 + 400 / (t+20)2 = 400 / (t+20)2

MTTF = integralinfinity0 t*f(t)dt

= integralinfinity0(400t / (t+20)2)dt

= infinity

With warranty period R(t) = 1- F(t) / (1 - F(1) )=1 - 1 + 400 / (t+20)2 = 400 / (t+20)2

Also t>1. Hence,

MTTF = integralinfinity1 t*f(t)dt

= integralinfinity1(400t / (t+20)2)dt

= infinity

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