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

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 k n j = − . [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
( )
1 2 3
function , ,...
k
Π = 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
n x
=
−
=
∑
, and usually expressed as a percentage.
Basic Statistics
•
Definitions – sample mean
1
1 n
i
i
x x
n
=
=
∑
, sample standard deviation
( ) 2
2
1 1
1 1
n n
i i
i i
d x
S
n n
= =
−
= =
− −
∑ ∑
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
x x
n x
=
⎛ ⎞
−
=
⎜
⎝ ⎠
∑
⎟
, 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.