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Unformatted text preview: Exam 1
ME 340 1. An engineer needs to measure velocity using a pitot tube with a pressure transducer, and has
been asked about the accuracy of the resulting measurements. The method is quite inexpensive,
but the engineer's boss is worried about the uncertainty in the result. The boss thinks that a
much more expensive measurement is needed. The expensive measurement system has a 2%
uncertainty (95% confidence) in velocity at the level of the measurement. The more expensive
method does not depend on the air density, p. The engineer prepares an uncertainty analysis of the pitot tube based on the data below. The question is: Should the engineer use the pitot tube
or the more expensive technique? Pitot + transducer characteristics: Full scale: 1” H20 Sensitivity error: 1% full scale Linearity error: 1% of reading Transducer resolution: 0.005” H20 50 data points were taken. Average Pressure: 0.5” H20, Standard Deviation of Readings: 0.01” H20 Data Reduction Equation: U = $where p is the density of the air being measured and Ap is the pressure measurement. The constant C is a conversion factor which converts the velocity into m/s for
the set—up being used. (a) What is the 95% confidence level uncertainty for (units of in H20):
a. Sensitivity ' b. Linearity c. Transducer resolution (b) What is the uncertainty in the measured pressure (not the average pressure)? ((2) What is the 95% statistical confidence interval ofthe average pressure? (cl) Find the uncertainty in the measured velocity (not average velocity) caused by the measurement
of pressure (assume C and p are both 1 for this part): (e) What is the 95% statistical confidence interval for the average velocity based on the average
pressure? (f) What other information or data would you like to know before answering the [3055’ question? '2. An engineer was attempting to fit some calibration data to a curve fit. She knew the data was
non—linear, but was not sure ofthe exact formulation. She attempted to fit a cubic polynomial to
the data using a technique taught in her statistics class. The attempted curve fit output and the plot of the output is given. Equation fit: 31 = a0 + alx + 51205 — DEF + a3(x — 37F SUMMA RY O UTPUT Regression Statistics Multiple R 0982414819
R Square 0365138878
Adjusted R Squal 0347708316
Standard Error 7.585038057
Observations 10
ANOVA df 55 MS F Significance F
Regression 3 9556859961 3185.619987 5537049924 9.14569E—05
Residual 6 345.1968139 5753280232
Total 9 9902056775 Coefficients Standard Error tStat Pvol'ue Lower 95% Upper 95% Intercept 8922093089 10.40399029 0857564535 0.424057893 3437974022 1653555405
x 11.11859579 1166790833 5.1313655?) 0.002154128 5816649623 1642054196
(xxmeany‘z 0914754467 0.330096411 2771173619 0032372228 0107037646 1722471288
(xxmean)"3 0.041671478 0.136478069 0305334611 0770426684 0.375621283 0292278327 Identify the coefficients for each term, ao, a1, a2, a3 Which coefficients are significantly different than zero? (support the answer) What is the 95% confidence interval on each coefficients? How many degrees of freedom went into the estimateof the error? What is the 95% confidence interval on the fit? Is the fit acceptable? If not, what should the engineer do next (don’t ask for more data)? 3. During an experiment, an engineer took 20 measurements of the length of wood parts being
used to construct furniture. The sample mean, 3? = 10, the sample variance, 5; = 0.25.
a. Construct a 95% Confidence interval for the mean b. Construct a 95% Confidence interval for the standard deviation. c. In this hypothetical example, the company can be fined if the average of any 20 sample
measurements is larger than 10.5, describe how you would determine the probability
that such a sample mean is larger than 10.5, based on the sample taken. Do not solve for the answer but set up the problem and write the equation. What distribution would
be used? 4K. Scientists studying the population of foxes and mice in a remote Indiana county took data on the
mice and fox populations monthly over a period of 20 years. A student processed the data using
an FFT and presented them with results below. (havens SPequom 5 a. What is the lowest and highest frequency present in the FT (not including the DC
component)? 20 la. What was the average fox population over the 20 year period? c. What is the primary PERIOD ofthe cycle in the fox population? d. What is the secondary PERIOD of the cycle in the fox population? e. Describe how you would use this data to determine the variance in the fox population
over the 20 year period. 576/. The figure below is the step response of a measurement system.
a. Describe the most likely system order (Zero, First or Second Order System) b. What is the ringing frequency of the measurement system? ’ c. If C = 0.3, what is the natural frequency of the system? d. Write the transfer function ofthe system e. Sketch the frequency response of the system (approximately). f. For the measurements needed, the response of the system is acceptable if the
measured value is within 25% of the actual values. What is the bandwidth of the system
FOR THESE MEASURMENTS? State the bandwidth in terms of frequencies. g. The engineer has a requirement to measure frequencies up to 100 hz,*can this
measurement system be used (using the criteria from part f)? Why or why not? ...
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 Fall '08
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