outline - Outline and Equation Sheet for M E 345 Author:...

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Outline and Equation Sheet for M E 345 Author: John M. Cimbala, Penn State University Latest revision, 03 January 2008 Introduction Primary dimensions – mass, length, time, temperature, current, amount of light, and amount of matter. Significant digits – the rules for multiplication and division, and for addition and subtraction. Rounding off round up if the least significant digit is odd , and truncate if the least significant digit is even . Dimensional Analysis Law of dimensional homogeneity Every additive term in an equation must have the same dimensions . The method of repeating variables – there are 6 steps: 1. List the parameters and count them, n . 2. List the primary dimensions of each parameter. 3. Set the reduction, j , as the number of primary dimensions in the problem. Then knj =− . [Reduce j by one if necessary.] 4. Choose j repeating variables. 5. Construct the k Π s, and manipulate as necessary. 6. Write out the final dimensionless functional relationship ( ) 12 3 function , ,... k Π Π Π and check your algebra. Dimensional analysis is often extremely useful in setting up and designing experiments. Errors and Calibration Systematic errors (bias errors) – consistent, repeatable errors. Random errors (precision errors) – scatter in data, a lack of repeatability, unrepeatable, inconsistent errors. Accuracy accuracy error is the measured value minus the true value . Precision precision error is the reading minus the average of readings . Other errors – zero, linearity, sensitivity, resolution, hysteresis, instrument reapeatability, drift. Calibration – static (time not relevant) vs. dynamic (time is relevant) calibration. Mean bias error – defined as true 1 true 1 MBE n i i x x nx = = , and usually expressed as a percentage. Basic Statistics Definitions – sample mean 1 1 n i i x x n = = , sample standard deviation () 2 2 11 nn ii dx S == −− ∑∑ x , sample variance ( S 2 ), sample median (half lower, half higher), sample mode (most probable value – one that occurs most frequently), population (all values) vs. sample (a selected portion of the total population). Excel – learn how to use Excel’s built-in statistics functions. Root mean square error – defined as 2 true 1 true 1 RMSE n i i xx = ⎛⎞ = ⎝⎠ , and usually expressed as a percentage. Histograms and Probability Density Functions Histogram plots – frequency, bins or classes, bin width or class width, Sturgis and Rice rules. Normalized histograms – how to normalize the vertical scale and the horizontal scale. PDF – how to create a probability density function from a histogram. Expected value – same as population mean. Standard deviation – same as population standard deviation.
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This note was uploaded on 04/05/2008 for the course ME 345 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.

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outline - Outline and Equation Sheet for M E 345 Author:...

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