lecture 6 mechanical measurement and instrumentation mecn 4600 department of mechanical engineering...

LectureLecture

66Mechanical Measurement and Mechanical Measurement and

InstrumentationInstrumentation MECN 4600 MECN 4600

Department of Mechanical EngineeringDepartment of Mechanical Engineering

Inter American University of Puerto RicoInter American University of Puerto Rico

Bayamon CampusBayamon Campus

Dr. Omar E. Meza CastilloDr. Omar E. Meza [email protected]

http://www.bc.inter.edu/facultad/omeza

Lecture 6Lecture 6MEC

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Tentative Lecture ScheduleTentative Lecture Schedule

TopicTopic LectureLecture

Basic Principles of MeasurementsBasic Principles of MeasurementsResponse of Measuring Systems, System DynamicsResponse of Measuring Systems, System DynamicsError & Uncertainty AnalysisError & Uncertainty Analysis

1, 2 and 31, 2 and 3

Least-Squares RegressionLeast-Squares Regression 44

Measurement of PressureMeasurement of Pressure 55

Measurement of TemperatureMeasurement of Temperature 66

Measurement of Fluid FlowMeasurement of Fluid Flow 77

Measurement of LevelMeasurement of Level 88

Measurement of Stress-StrainMeasurement of Stress-Strain 99

Measurement of Time ConstantMeasurement of Time Constant 1010


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Statistics Theory– CalibrationStatistics Theory– Calibration

Pressure Measurements using a Pressure Measurements using a Bourdon GaugeBourdon Gauge

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To learn statistics techniques for To learn statistics techniques for calibration of a Bourdon Gage. calibration of a Bourdon Gage.

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Course ObjectivesCourse Objectives


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on Statistics TheoryStatistics Theory

Statistics: Statistics: are mathematical tools used to are mathematical tools used to organize, summarize, and manipulate organize, summarize, and manipulate data.data.

Data: Data: The measurements obtained in a The measurements obtained in a research study are called the data. The research study are called the data. The goal of statistics is to help researchers goal of statistics is to help researchers organize and interpret the data. organize and interpret the data. Information expressed as numbers Information expressed as numbers (quantitatively).(quantitatively).

Variable: Variable: Traits that can change values Traits that can change values from case to case. Examples:from case to case. Examples: Weight, Temperature, Level, Pressure, etc.Weight, Temperature, Level, Pressure, etc.

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on Types Of VariablesTypes Of Variables

Variables may be:Variables may be: Independent or dependentIndependent or dependent

In causal relationships:In causal relationships: CAUSE CAUSE EFFECT EFFECT

independent variable independent variable dependent variable dependent variable Discrete or continuousDiscrete or continuous

Discrete variables are measured in units Discrete variables are measured in units that cannot be subdivided.that cannot be subdivided. Example: Number of childrenExample: Number of children

Continuous variables are measured in a Continuous variables are measured in a unit that can be subdivided infinitely.unit that can be subdivided infinitely. Example: AgeExample: Age

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on Descriptive StatisticsDescriptive Statistics

Descriptive statistics Descriptive statistics are methods for are methods for organizing and summarizing data. organizing and summarizing data.

For example, tables or graphs are used to For example, tables or graphs are used to organize data, and descriptive values organize data, and descriptive values such as the average score are used to such as the average score are used to summarize data. summarize data.

A descriptive value for a population is A descriptive value for a population is called a parameter and a descriptive called a parameter and a descriptive value for a sample is called a statistic. value for a sample is called a statistic.

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on Inferential StatisticsInferential Statistics

Inferential statistics Inferential statistics are methods for are methods for using sample data to make general using sample data to make general conclusions (inferences) about conclusions (inferences) about populations. populations.

Because a sample is typically only a part Because a sample is typically only a part of the whole population, sample data of the whole population, sample data provide only limited information about provide only limited information about the population. As a result, sample the population. As a result, sample statistics are generally imperfect statistics are generally imperfect representatives of the corresponding representatives of the corresponding population parameters. population parameters.

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Nominal, ordinal, or interval-ratioNominal, ordinal, or interval-ratio Nominal - Scores are labels only, they are Nominal - Scores are labels only, they are

not numbers.not numbers. Ordinal - Scores have some numerical Ordinal - Scores have some numerical

quality and can be ranked.quality and can be ranked. Interval-ratio - Scores are numbers Interval-ratio - Scores are numbers

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on Frequency DistributionsFrequency Distributions

After collecting data, the first task for a After collecting data, the first task for a researcher is to researcher is to organizeorganize and and simplifysimplify the the data so that it is possible to get a general data so that it is possible to get a general overview of the results. overview of the results.

This is the goal of descriptive statistical This is the goal of descriptive statistical techniques. techniques.

One method for simplifying and One method for simplifying and organizing data is to construct a organizing data is to construct a frequency distributionfrequency distribution. .

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on Frequency Distributions (cont.)Frequency Distributions (cont.)

A A frequency distribution frequency distribution is an organized is an organized tabulation showing exactly how many tabulation showing exactly how many individuals are located in each category individuals are located in each category on the scale of measurement. A on the scale of measurement. A frequency distribution presents an frequency distribution presents an organized picture of the entire set of organized picture of the entire set of scores, and it shows where each scores, and it shows where each individual is located relative to others in individual is located relative to others in the distribution. the distribution.

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on Frequency Distribution TablesFrequency Distribution Tables

A A frequency distribution frequency distribution table consists of table consists of at least two columns - one listing at least two columns - one listing categories on the scale of measurement categories on the scale of measurement (X) and another for frequency (f). (X) and another for frequency (f).

In the X column, values are listed from In the X column, values are listed from the highest to lowest, without skipping the highest to lowest, without skipping any. any.

For the frequency column, tallies are For the frequency column, tallies are determined for each value (how often determined for each value (how often each X value occurs in the data set). each X value occurs in the data set). These tallies are the frequencies for each These tallies are the frequencies for each X value. X value.

The sum of the frequencies should equal The sum of the frequencies should equal N. N.

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on Frequency Distribution Tables (cont.)Frequency Distribution Tables (cont.)

A third column can be used for the A third column can be used for the proportion (p) for each category: p = f/N. proportion (p) for each category: p = f/N. The sum of the p column should equal The sum of the p column should equal 1.00. 1.00.

A fourth column can display the A fourth column can display the percentage of the distribution percentage of the distribution corresponding to each X value. The corresponding to each X value. The percentage is found by multiplying p by percentage is found by multiplying p by 100. The sum of the percentage column 100. The sum of the percentage column is 100%. is 100%.

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on Regular and Grouped Frequency DistributionRegular and Grouped Frequency Distribution

When a frequency distribution table lists When a frequency distribution table lists all of the individual categories (X values) all of the individual categories (X values) it is called a it is called a regular frequency regular frequency distributiondistribution. .

Sometimes, however, a set of scores Sometimes, however, a set of scores covers a wide range of values. In these covers a wide range of values. In these situations, a list of all the X values would situations, a list of all the X values would be quite long - too long to be a “simple” be quite long - too long to be a “simple” presentation of the data. presentation of the data.

To remedy this situation, a To remedy this situation, a grouped grouped frequency distributionfrequency distribution table is used, table is used, where the X column lists groups of scores, where the X column lists groups of scores, called called class intervalsclass intervals..

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on Frequency Distribution GraphsFrequency Distribution Graphs

In a In a frequency distribution graphfrequency distribution graph, the , the score categories (X values) are listed on score categories (X values) are listed on the X axis and the frequencies are listed the X axis and the frequencies are listed on the Y axis. on the Y axis.

When the score categories consist of When the score categories consist of numerical scores from an interval or ratio numerical scores from an interval or ratio scale, the graph should be either a scale, the graph should be either a histogram or a polygonhistogram or a polygon. .

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on HistogramsHistograms

In a In a histogramhistogram, a bar is centered above , a bar is centered above each score (or class interval) so that the each score (or class interval) so that the height of the bar corresponds to the height of the bar corresponds to the frequency and the width extends to the frequency and the width extends to the real limits, so that adjacent bars touch. real limits, so that adjacent bars touch.

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HistogramsHistograms


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PolygonsPolygons

In a In a polygonpolygon, a dot is centered above each , a dot is centered above each score so that the height of the dot score so that the height of the dot corresponds to the frequency. The dots corresponds to the frequency. The dots are then connected by straight lines. An are then connected by straight lines. An additional line is drawn at each end to additional line is drawn at each end to bring the graph back to a zero frequency. bring the graph back to a zero frequency.


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Smooth curveSmooth curve

If the scores in the population are If the scores in the population are measured on an interval or ratio scale, it measured on an interval or ratio scale, it is customary to present the distribution is customary to present the distribution as a smooth curve rather than a jagged as a smooth curve rather than a jagged histogram or polygon. histogram or polygon.

The smooth curve emphasizes the fact The smooth curve emphasizes the fact that the distribution is not showing the that the distribution is not showing the exact frequency for each category.exact frequency for each category.


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Frequency distribution graphsFrequency distribution graphs

Frequency distribution graphs are useful Frequency distribution graphs are useful because they show the entire set of because they show the entire set of scores. scores.

At a glance, you can determine the At a glance, you can determine the highest score, the lowest score, and highest score, the lowest score, and where the scores are centered. where the scores are centered.

The graph also shows whether the scores The graph also shows whether the scores are clustered together or scattered over a are clustered together or scattered over a wide range. wide range.


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ShapeShape

A graph shows the shape of the A graph shows the shape of the distribution. distribution.

A distribution is symmetrical if the left A distribution is symmetrical if the left side of the graph is (roughly) a mirror side of the graph is (roughly) a mirror image of the right side. image of the right side.

One example of a symmetrical One example of a symmetrical distribution is the distribution is the bell-shaped normal bell-shaped normal distributiondistribution. .

On the other hand, distributions are On the other hand, distributions are skewed when scores pile up on one side skewed when scores pile up on one side of the distribution, leaving a "tail" of a of the distribution, leaving a "tail" of a few extreme values on the other side. few extreme values on the other side.


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on Positively and Negatively Skewed DistributionsPositively and Negatively Skewed Distributions

In a In a positively skewed distributionpositively skewed distribution, the , the scores tend to pile up on the left side of scores tend to pile up on the left side of the distribution with the tail tapering off the distribution with the tail tapering off to the right. to the right.

In a In a negatively skewed distributionnegatively skewed distribution, the , the scores tend to pile up on the right side scores tend to pile up on the right side and the tail points to the left.and the tail points to the left.

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on Positively and Negatively Skewed DistributionsPositively and Negatively Skewed Distributions


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Step by StepStep by Step

Como Construir un HistogramaComo Construir un Histograma

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on Data para construir un HistogramaData para construir un Histograma

Debe abrir una hoja de ExcelDebe abrir una hoja de Excel Verificar si tiene disponible la Verificar si tiene disponible la

herramienta Análisis de datos herramienta Análisis de datos Si posee la herramienta, deberá introducir Si posee la herramienta, deberá introducir

los siguientes datos en la hoja de Excellos siguientes datos en la hoja de Excel

0.19 0.19 0.37 0.270.35 0.17 0.22 0.320.37 0.2 0.27 0.390.25 0.3 0.27 0.370.29 0.32 0.26 0.320.22 0.24 0.29 0.260.32 0.32 0.27 0.270.27 0.34 0.22 0.280.27 0.15 0.23 0.280.22 0.27 0.26 0.27


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on Pasos a pasoPasos a paso

1.1. Calcular valor máx. y min.Calcular valor máx. y min. 2.2. Calcular la diferencia entre máx. y min. Calcular la diferencia entre máx. y min.

R= rango.R= rango.3.3. Calcular el numero de clases Calcular el numero de clases

h=1+3.32*log(n).h=1+3.32*log(n).4.4. Calcular ancho de clase C= R/h.Calcular ancho de clase C= R/h.5.5. Construya el histograma con los datos Construya el histograma con los datos

obtenidos.obtenidos.6.6. Con el modelo que se te proporciona a Con el modelo que se te proporciona a

continuación podrás resolver tu continuación podrás resolver tu problema. problema.


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on Introducir los Datos en la Hoja de Excel Introducir los Datos en la Hoja de Excel

Esta es la Celda A1


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on Entonces tu hoja se vera así: Entonces tu hoja se vera así:


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on Calcular los valores Calcular los valores MáximoMáximo y Mínimo y Mínimo

En la celda B14 escribes la formula

y le das Enter


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on Calcular los valores Máximo y Calcular los valores Máximo y MínimoMínimo

En la celda B15 escribes la formula

y le das Enter


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on Calcular del Calcular del Número de datosNúmero de datos

En la celda B16 escribes la formula y

le das Enter


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on Calcular el Calcular el RangoRango

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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on Calcular el Calcular el Número de clasesNúmero de clases

Ahora tienes que introducir la formula como aparece en la celda A18 y le das

Enter


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on Calcular el Calcular el Ancho de claseAncho de clase

Aquí escribes (=) y relacionas las celdas B17/B18 y dale Enter


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on Construir las Construir las ClasesClases (Min. y Max. Del intervalo) (Min. y Max. Del intervalo)

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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on Menú Herramientas y selecciona Análisis de datosMenú Herramientas y selecciona Análisis de datos


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on En la ventana seleccionar la opción En la ventana seleccionar la opción histogramahistograma


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on Se presentara la siguiente ventana Se presentara la siguiente ventana


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on Seleccionar Seleccionar A1:D10A1:D10 en Rango de entrada en Rango de entrada


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on Seleccionar Seleccionar B22:B27B22:B27 en Rango de clase en Rango de clase


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on Seleccionar Seleccionar D19D19 para Rango de salida y Crear Grafico para Rango de salida y Crear Grafico


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on La tabla se inicia en la celda La tabla se inicia en la celda D19D19


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on Si le efectuamos modificaciones al grafico se vera asíSi le efectuamos modificaciones al grafico se vera así


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on Measuring the Measuring the MeanMean

Notation: Notation: It is simply the ordinary arithmetic It is simply the ordinary arithmetic

average.average. Suppose that we have n observations Suppose that we have n observations

(data size, number of individuals).(data size, number of individuals). Observations are denoted as xObservations are denoted as x11, x, x22, x, x33, …, …

xxnn.. How to get ? How to get ?

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x

x

n

x

n

x...xxxx in321


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on Measuring the Measuring the Standard DeviationStandard Deviation

It says how far the observations are from It says how far the observations are from their mean. The their mean. The variancevariance s s22 of a set of of a set of observations is an average of the squares observations is an average of the squares of the deviations of the observations from of the deviations of the observations from their mean. their mean.

Notation: sNotation: s22 for for variancevariance and s for and s for standard deviation standard deviation

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1n

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1n

)xx(s

1n

)xx(...)xx()xx(s

22i

2i2

2n

22

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Bourdon GaugeBourdon Gauge

PressurePressure

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on PressurePressure

For a For a fluid at restfluid at rest, pressure is the same in , pressure is the same in all directions all directions at this point. But can vary at this point. But can vary from point to point, e.g. hydrostatic from point to point, e.g. hydrostatic pressure. P=F/Apressure. P=F/A

For a For a fluid in motion fluid in motion additional forces arise additional forces arise due to shearing action and we refer to the due to shearing action and we refer to the normal force as a normal stress. The state normal force as a normal stress. The state of stresses in a fluid in motion is dealt with of stresses in a fluid in motion is dealt with further in further in Fluid Mechanics.Fluid Mechanics.

In the context of In the context of thermodynamicsthermodynamics, we think , we think of pressure as of pressure as absolute, with respect to absolute, with respect to pressure of a complete pressure of a complete vacuum (space) vacuum (space) which is zero.which is zero.

In In Fluid Mechanics Fluid Mechanics we often use gage we often use gage pressure and vacuum pressure.pressure and vacuum pressure.

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on PressurePressure

Absolute PressureAbsolute PressureForce per unit area Force per unit area exerted by a fluidexerted by a fluid

Gage PressureGage PressurePressure abovePressure above

atmosphericatmospheric

PPgagegage= P= Pabsabs - P - Patmatm

Vacuum PressureVacuum PressurePressure below Pressure below

atmosphericatmospheric

PPvacvac=-P=-Pgagegage= P= Patmatm - P - Pabsabs

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on PressurePressure

Common Pressure Units are:Common Pressure Units are:

Pa (Pascal), mmHg (mm of Mercury), atm Pa (Pascal), mmHg (mm of Mercury), atm (atmosphere), psi (lbf per square inch)(atmosphere), psi (lbf per square inch)

1 Pa = 1 N/m1 Pa = 1 N/m22 (S.I. Unit)(S.I. Unit) 1 kPa =101 kPa =1033 PaPa 1 bar = 101 bar = 1055 Pa (note the bar is not an SI Pa (note the bar is not an SI

unit)unit) 1 MPa = 101 MPa = 1066 PaPa

1 atm = 760 mmHg = 101,325 Pa = 14.696 1 atm = 760 mmHg = 101,325 Pa = 14.696 psipsi

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on Variation of Pressure with DepthVariation of Pressure with Depth

If we take point 1 to be at If we take point 1 to be at the free surface of a liquid the free surface of a liquid open to the atmosphere, open to the atmosphere, where the pressure is the where the pressure is the atmospheric pressure Patmospheric pressure Patmatm, , then the pressure at a depth then the pressure at a depth hh from the free surface from the free surface becomesbecomes

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on ManometerManometer

Many engineering problems and some manometers Many engineering problems and some manometers involve multiple immiscible fluids of different involve multiple immiscible fluids of different densities stacked on top of each other. Such systems densities stacked on top of each other. Such systems can be analyzed easily by remembering that:can be analyzed easily by remembering that:

1.1. The pressure change across a fluid column of The pressure change across a fluid column of height h is height h is ΔΔP=P=ρρghgh

2.2. Pressure increases downward in a given fluid Pressure increases downward in a given fluid and decreases upward (i.e., Pand decreases upward (i.e., Pbottombottom>P>Ptoptop), and ), and

3.3. Two points at the same elevation in a Two points at the same elevation in a continuous fluid at rest are at the same continuous fluid at rest are at the same pressure.pressure.

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on ManometerManometer

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on Pressure - Bourdon Tube GaugePressure - Bourdon Tube Gauge

A Bourdon gauge uses a coiled tube which as it A Bourdon gauge uses a coiled tube which as it expands due to pressure increase causes a expands due to pressure increase causes a rotation of an arm connected to the tuberotation of an arm connected to the tube

The pressure sensing element is a closed coiled The pressure sensing element is a closed coiled tube connected to the chamber or pipe in which tube connected to the chamber or pipe in which pressure is to be sensedpressure is to be sensed

As the gauge pressure increases the tube will tend As the gauge pressure increases the tube will tend to uncoil, while a reduced gauge pressure will to uncoil, while a reduced gauge pressure will cause the tube to coil more tightlycause the tube to coil more tightly

This motion is transferred through a linkage to a This motion is transferred through a linkage to a gear train connected to an indicating needle. The gear train connected to an indicating needle. The needle is presented in front of a card face needle is presented in front of a card face inscribed with the pressure indications associated inscribed with the pressure indications associated with particular needle deflectionswith particular needle deflections

Note that a Bourdon gauge can measure liquid Note that a Bourdon gauge can measure liquid pressure as well as gas pressurepressure as well as gas pressure


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5656

Pressure - Bourdon Tube GaugePressure - Bourdon Tube Gauge


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on Pressure - Bourdon Tube gaugePressure - Bourdon Tube gauge


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on Pressure – Dead-Weight TesterPressure – Dead-Weight Tester

Pressure transducers can be recalibrated Pressure transducers can be recalibrated on-line or in a calibration laboratoryon-line or in a calibration laboratory

Laboratory recalibration typically is Laboratory recalibration typically is preferred, but often is not possible or preferred, but often is not possible or necessarynecessary

In the laboratory, there usually are two In the laboratory, there usually are two types of calibration devices: deadweight types of calibration devices: deadweight testers that provide primary, base-line testers that provide primary, base-line standards, and "laboratory" or "field" standards, and "laboratory" or "field" standard calibration devices that are standard calibration devices that are periodically recalibrated against the periodically recalibrated against the primaryprimary

Of course, these secondary standards are Of course, these secondary standards are less accurate than the primary, but they less accurate than the primary, but they provide a more convenient means of provide a more convenient means of testing other instruments. testing other instruments.


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on Pressure – Dead-Weight TesterPressure – Dead-Weight Tester

A deadweight tester A deadweight tester consists of a pumping consists of a pumping piston with a screw that piston with a screw that presses it into the presses it into the reservoir, a primary piston reservoir, a primary piston that carries the dead that carries the dead weight, and the gauge or weight, and the gauge or transducer to be tested transducer to be tested

It works by loading the It works by loading the primary piston (of cross primary piston (of cross sectional area A), with the sectional area A), with the amount of weight (W) that amount of weight (W) that corresponds to the desired corresponds to the desired calibration pressure (P = calibration pressure (P = W/A)W/A)

Bourdon Gage

Dead Weight tester


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Bourdon GaugeBourdon Gauge

CalibrationCalibration

6060


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on CalibrationCalibration

The proper for calibration is to apply The proper for calibration is to apply known inputs ranging from the minimum known inputs ranging from the minimum to maximum values for which the to maximum values for which the measurement system is to be used.measurement system is to be used.

These limits define the operating These limits define the operating RANGERANGE of the system. The input operating range of the system. The input operating range is defined as extending from xis defined as extending from xminmin to x to xmaxmax. . This range defines its This range defines its INPUT SPANINPUT SPAN, , expressed asexpressed as

6161


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Similarly, the output operating range is Similarly, the output operating range is specific from yspecific from yminmin to y to ymaxmax. The . The OUT SPANOUT SPAN, , or or FULL-SCALE OPERATING RANGE (FSO)FULL-SCALE OPERATING RANGE (FSO), , is expressed asis expressed as

ACCURACY: ACCURACY: The accuracy of a The accuracy of a measurement system refers to its ability measurement system refers to its ability to indicate a true value exactly. Accuracy to indicate a true value exactly. Accuracy is related to absolute error.is related to absolute error.

6262


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Absolute error Absolute error εε, is defined as the , is defined as the difference between the true value applied difference between the true value applied to a measurement system and the to a measurement system and the indicated value of the system.indicated value of the system.

6363


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Precision ErrorPrecision Error: The precision error is a : The precision error is a measure of the random variation found measure of the random variation found during repeated measurements.during repeated measurements.

Bias ErrorBias Error: The bias error is the difference : The bias error is the difference between the average value and the true between the average value and the true value.value.

6464


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on Hysteresis ErrorHysteresis Error

Refers to differences in the values found Refers to differences in the values found between going upscale and downscale in between going upscale and downscale in a sequential test.a sequential test.

HysteresisHysteresis is usually specified for a is usually specified for a measurement system in terms of the measurement system in terms of the maximum hysteresis error found in the maximum hysteresis error found in the calibration, ecalibration, ehhmaxmax

, as a percentage of full-, as a percentage of full-

scale output range:scale output range:

6565


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6666


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on Linearity ErrorLinearity Error

Refers to differences found between Refers to differences found between measured value y(x) and the curve fit measured value y(x) and the curve fit yyLL(x):(x):

For a measurement device is often For a measurement device is often specified in terms of the maximum specified in terms of the maximum expected linearity error for the calibration expected linearity error for the calibration as a percentage of full-scale output as a percentage of full-scale output range:range:

6767


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6868


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on Sensitivity ShiftSensitivity Shift

6969


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7070


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7171


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on Zero ShiftZero Shift

7272


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7373


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7474


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7575


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7676


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7777


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on Sensitivity and Zero ShiftSensitivity and Zero Shift

7878


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7979


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8080


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Due, Due, Wednesday, February 02, 2011Wednesday, February 02, 2011

Omar E. Meza Castillo Ph.D.Omar E. Meza Castillo Ph.D.

Homework1 Homework1 WebPage WebPage

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lecture 6 mechanical measurement and instrumentation mecn 4600 department of mechanical engineering...

Documents

sample statistics

sample data

measurement of pressure

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goal of statistics

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