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
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.