Transcript 6 Mechanical Measurement and Instrumentation MECN 4600

Lecture
6
Mechanical Measurement and
Instrumentation
MECN 4600
Department of Mechanical Engineering
Inter American University of Puerto Rico
Bayamon Campus
Dr. Omar E. Meza Castillo
[email protected]
http://www.bc.inter.edu/facultad/omeza
Inter - Bayamon
Tentative Lecture Schedule
Topic
Lecture
Basic Principles of Measurements
Response of Measuring Systems, System Dynamics
Error & Uncertainty Analysis
Least-Squares Regression
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Measurement of Pressure
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Measurement of Temperature
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Measurement of Fluid Flow
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Measurement of Level
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Measurement of Stress-Strain
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Measurement of Time Constant
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1, 2 and 3
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Pressure Measurements using a
Bourdon Gauge
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Statistics Theory– Calibration
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Course Objectives
 To
learn
statistics
techniques
calibration of a Bourdon Gage.
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Statistics Theory
 Statistics: are mathematical tools used to
organize, summarize, and manipulate
data.
 Data: The measurements obtained in a
research study are called the data. The
goal of statistics is to help researchers
organize
and
interpret
the
data.
Information
expressed
as
numbers
(quantitatively).
 Variable: Traits that can change values
from case to case. Examples:
 Weight, Temperature, Level, Pressure, etc.
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Types Of Variables
 Variables may be:
 Independent or dependent
 In causal relationships:
CAUSE

EFFECT
independent variable  dependent variable
 Discrete or continuous
 Discrete variables are measured in units
that cannot be subdivided.
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 Example: Number of children
 Continuous variables are measured in a
unit that can be subdivided infinitely.
 Example: Age
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Descriptive Statistics
 Descriptive statistics are methods for
organizing and summarizing data.
 For example, tables or graphs are used to
organize data, and descriptive values such
as the average score are used to
summarize data.
 A descriptive value for a population is
called a parameter and a descriptive value
for a sample is called a statistic.
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Inferential Statistics
 Inferential statistics are methods for
using sample data to make general
conclusions
(inferences)
about
populations.
 Because a sample is typically only a part
of the whole population, sample data
provide only limited information about the
population. As a result, sample statistics
are generally imperfect representatives of
the corresponding population parameters.
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Types Of Variables
 Nominal, ordinal, or interval-ratio
 Nominal - Scores are labels only, they are
not numbers.
 Ordinal - Scores have some numerical
quality and can be ranked.
 Interval-ratio - Scores are numbers
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Frequency Distributions
 After collecting data, the first task for a
researcher is to organize and simplify the
data so that it is possible to get a general
overview of the results.
 This is the goal of descriptive statistical
techniques.
 One method for simplifying and organizing
data
is
to
construct
a
frequency
distribution.
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Frequency Distributions (cont.)
 A frequency distribution is an organized
tabulation showing exactly how many
individuals are located in each category on
the scale of measurement. A frequency
distribution presents an organized picture
of the entire set of scores, and it shows
where each individual is located relative
to others in the distribution.
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Frequency Distribution Tables
 A frequency distribution table consists of
at least two columns - one listing
categories on the scale of measurement
(X) and another for frequency (f).
 In the X column, values are listed from
the highest to lowest, without skipping
any.
 For the frequency column, tallies are
determined for each value (how often
each X value occurs in the data set).
These tallies are the frequencies for each
X value.
 The sum of the frequencies should equal
N.
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Frequency Distribution Tables (cont.)
 A third column can be used for the
proportion (p) for each category: p = f/N.
The sum of the p column should equal
1.00.
 A
fourth
column
can
display
the
percentage
of
the
distribution
corresponding to each X value.
The
percentage is found by multiplying p by
100. The sum of the percentage column is
100%.
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Regular and Grouped Frequency Distribution
 When a frequency distribution table lists
all of the individual categories (X values)
it
is
called
a
regular
frequency
distribution.
 Sometimes, however, a set of scores
covers a wide range of values. In these
situations, a list of all the X values would
be quite long - too long to be a “simple”
presentation of the data.
 To remedy this situation, a grouped
frequency distribution table is used,
where the X column lists groups of scores,
called class intervals.
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Frequency Distribution Graphs
 In a frequency distribution graph, the
score categories (X values) are listed on
the X axis and the frequencies are listed
on the Y axis.
 When the score categories consist of
numerical scores from an interval or ratio
scale, the graph should be either a
histogram or a polygon.
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Histograms
 In a histogram, a bar is centered above
each score (or class interval) so that the
height of the bar corresponds to the
frequency and the width extends to the
real limits, so that adjacent bars touch.
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Histograms
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Polygons
 In a polygon, a dot is centered above each
score so that the height of the dot
corresponds to the frequency. The dots
are then connected by straight lines. An
additional line is drawn at each end to
bring the graph back to a zero frequency.
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Polygons
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Smooth curve
 If the scores in the population are
measured on an interval or ratio scale, it
is customary to present the distribution as
a smooth curve rather than a jagged
histogram or polygon.
 The smooth curve emphasizes the fact
that the distribution is not showing the
exact frequency for each category.
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Frequency distribution graphs
 Frequency distribution graphs are useful
because they show the entire set of
scores.
 At a glance, you can determine the
highest score, the lowest score, and
where the scores are centered.
 The graph also shows whether the scores
are clustered together or scattered over a
wide range.
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Shape
 A graph shows the shape of the
distribution.
 A distribution is symmetrical if the left
side of the graph is (roughly) a mirror
image of the right side.
 One example of a symmetrical distribution
is the bell-shaped normal distribution.
 On the other hand, distributions are
skewed when scores pile up on one side of
the distribution, leaving a "tail" of a few
extreme values on the other side.
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Positively and Negatively Skewed Distributions
 In a positively skewed distribution, the
scores tend to pile up on the left side of
the distribution with the tail tapering off
to the right.
 In a negatively skewed distribution, the
scores tend to pile up on the right side
and the tail points to the left.
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Positively and Negatively Skewed Distributions
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Como Construir un Histograma
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Step by Step
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Data para construir un Histograma
 Debe abrir una hoja de Excel
 Verificar si tiene disponible la herramienta
Análisis de datos
 Si posee la herramienta, deberá introducir
los siguientes datos en la hoja de Excel
0.19
0.35
0.37
0.25
0.29
0.22
0.32
0.27
0.27
0.22
0.19
0.17
0.2
0.3
0.32
0.24
0.32
0.34
0.15
0.27
0.37
0.22
0.27
0.27
0.26
0.29
0.27
0.22
0.23
0.26
0.27
0.32
0.39
0.37
0.32
0.26
0.27
0.28
0.28
0.27
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Pasos a paso
1. Calcular valor máx. y min.
2. Calcular la diferencia entre máx. y min.
R= rango.
3. Calcular
el
numero
de
clases
h=1+3.32*log(n).
4. Calcular ancho de clase C= R/h.
5. Construya el histograma con los datos
obtenidos.
6. Con el modelo que se te proporciona a
continuación
podrás
resolver
tu
problema.
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Introducir los Datos en la Hoja de Excel
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Esta es la Celda A1
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Entonces tu hoja se vera así:
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Calcular los valores Máximo y Mínimo
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En la celda B14
escribes la formula
y le das Enter
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Calcular los valores Máximo y Mínimo
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En la celda B15
escribes la formula
y le das Enter
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Calcular del Número de datos
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En la celda B16
escribes la formula
y le das Enter
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Calcular el Rango
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En esta celda escribes el
(=) y marcas la celda
B14 Luego digitas el (–)
y marcas la celda B15 y
le das Enter
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Calcular el Número de clases
Ahora tienes que
introducir la formula
como aparece en la
celda A18 y le das
Enter
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Calcular el Ancho de clase
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Aquí escribes (=) y
relacionas las celdas
B17/B18 y dale Enter
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Construir las Clases (Min. y Max. Del intervalo)
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Para construir la tabla sumaras
el valor de la celda B15+B19
y así sucesivamente hasta
obtener el valor de la celda B27
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Menú Herramientas y selecciona Análisis de datos
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En la ventana seleccionar la opción histograma
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Se presentara la siguiente ventana
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Seleccionar A1:D10 en Rango de entrada
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Seleccionar B22:B27 en Rango de clase
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Seleccionar D19 para Rango de salida y Crear Grafico
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La tabla se inicia en la celda D19
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Si le efectuamos modificaciones al grafico se vera así
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Measuring the Mean
 Notation: x
 It is simply the ordinary arithmetic
average.
 Suppose that we have n observations
(data size, number of individuals).
 Observations are denoted as x1, x2, x3,
…xn.
 How to get x ?
x1  x 2  x 3  ...  x n
x

n
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x
i
n
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Measuring the Standard Deviation
 It says how far the observations are from
their mean. The variance s2 of a set of
observations is an average of the squares
of the deviations of the observations from
their mean.
 Notation: s2 for variance and s for
standard deviation
( x1  x ) 2  ( x 2  x ) 2  ...  ( x n  x ) 2
s 
n 1
2
(
x

x
)

i
s2 
n 1
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2
s
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( x i2 )  n ( x ) 2
n 1
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Pressure
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Bourdon Gauge
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Pressure
 For a fluid at rest, pressure is the same in
all directions at this point. But can vary
from point to point, e.g. hydrostatic
pressure. P=F/A
 For a fluid in motion additional forces arise
due to shearing action and we refer to the
normal force as a normal stress. The state
of stresses in a fluid in motion is dealt with
further in Fluid Mechanics.
 In the context of thermodynamics, we
think of pressure as absolute, with respect
to pressure of a complete vacuum (space)
which is zero.
 In Fluid Mechanics we often use gage
pressure and vacuum pressure.
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Pressure
 Absolute Pressure
Force
per
unit
exerted by a fluid
area
 Gage Pressure
Pressure above
atmospheric
Pgage= Pabs - Patm
 Vacuum Pressure
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Pressure
atmospheric
Pvac=-Pgage=
Pabs
below
Patm
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Pressure
 Common Pressure Units are:
Pa (Pascal), mmHg (mm of Mercury), atm
(atmosphere), psi (lbf per square inch)
 1 Pa = 1 N/m2 (S.I. Unit)
 1 kPa =103 Pa
 1 bar = 105 Pa (note the bar is not an SI
unit)
 1 MPa = 106 Pa
 1 atm = 760 mmHg = 101,325 Pa = 14.696
psi
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Variation of Pressure with Depth
If we take point 1 to be at
the free surface of a liquid
open to the atmosphere,
where the pressure is the
atmospheric pressure Patm,
then the pressure at a depth
h from the free surface
becomes
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Manometer
Many engineering problems and some manometers
involve multiple immiscible fluids of different
densities stacked on top of each other. Such systems
can be analyzed easily by remembering that:
1. The pressure change across a fluid column of
height h is ΔP=ρgh
2. Pressure increases downward in a given fluid
and decreases upward (i.e., Pbottom>Ptop), and
3. Two points at the same elevation in a continuous
fluid at rest are at the same pressure.
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Manometer
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Pressure - Bourdon Tube Gauge
 A Bourdon gauge uses a coiled tube which as it
expands due to pressure increase causes a
rotation of an arm connected to the tube
 The pressure sensing element is a closed coiled
tube connected to the chamber or pipe in which
pressure is to be sensed
 As the gauge pressure increases the tube will tend
to uncoil, while a reduced gauge pressure will
cause the tube to coil more tightly
 This motion is transferred through a linkage to a
gear train connected to an indicating needle. The
needle is presented in front of a card face
inscribed with the pressure indications associated
with particular needle deflections
 Note that a Bourdon gauge can measure liquid
pressure as well as gas pressure
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Pressure - Bourdon Tube Gauge
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Pressure - Bourdon Tube gauge
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Pressure – Dead-Weight Tester
 Pressure transducers can be recalibrated
on-line or in a calibration laboratory
 Laboratory
recalibration
typically
is
preferred, but often is not possible or
necessary
 In the laboratory, there usually are two
types of calibration devices: deadweight
testers that provide primary, base-line
standards, and "laboratory" or "field"
standard calibration devices that are
periodically
recalibrated
against
the
primary
 Of course, these secondary standards are
less accurate than the primary, but they
provide a more convenient means of
testing other instruments.
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Pressure – Dead-Weight Tester
 A
deadweight
tester
consists of a pumping
piston with a screw that
presses
it
into
the
reservoir,
a
primary
piston that carries the
dead weight, and the
gauge or transducer to
be tested
 It works by loading the
primary piston (of cross
sectional area A), with
the amount of weight
(W) that corresponds to
the desired calibration
pressure (P = W/A)
Dead Weight tester
Bourdon Gage
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Calibration
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Bourdon Gauge
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Calibration
 The proper for calibration is to apply
known inputs ranging from the minimum
to maximum values for which the
measurement system is to be used.
 These limits define the operating RANGE
of the system. The input operating range
is defined as extending from xmin to xmax.
This range defines its INPUT SPAN,
expressed as
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Calibration
 Similarly, the output operating range is
specific from ymin to ymax. The OUT SPAN,
or FULL-SCALE OPERATING RANGE (FSO),
is expressed as
 ACCURACY:
The
accuracy
of
a
measurement system refers to its ability
to indicate a true value exactly. Accuracy
is related to absolute error.
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Calibration
 Absolute error
ε, is defined as the
difference between the true value applied
to a measurement system and the
indicated value of the system.
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Calibration
 Precision Error: The precision error is a
measure of the random variation found
during repeated measurements.
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 Bias Error: The bias error is the difference
between the average value and the true
value.
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Hysteresis Error
 Refers to differences in the values found
between going upscale and downscale in a
sequential test.
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 Hysteresis is usually specified for a
measurement system in terms of the
maximum hysteresis error found in the
calibration, ehmax, as a percentage of fullscale output range:
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Hysteresis Error
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Linearity Error
 Refers to differences found between
measured value y(x) and the curve fit
yL(x):
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 For a measurement device is often
specified in terms of the maximum
expected linearity error for the calibration
as a percentage of full-scale output range:
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Linearity Error
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Sensitivity Shift
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Sensitivity Shift
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Sensitivity Shift
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Zero Shift
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Zero Shift
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Zero Shift
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Hysteresis Error
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Linearity Error
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Sensitivity and Zero Shift
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Sensitivity and Zero Shift
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Sensitivity and Zero Shift
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Homework1  WebPage
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Due, Wednesday, February 02, 2011
Omar E. Meza Castillo Ph.D.
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