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**Mechanical Measurement and Instrumentation MECN 4600**

Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus Dr. Omar E. Meza Castillo

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**Tentative Lecture Schedule**

Topic Lecture Basic Principles of Measurements Response of Measuring Systems, System Dynamics Error & Uncertainty Analysis 1, 2 and 3 Least-Squares Regression 4 Measurement of Pressure 5 Measurement of Temperature 6 Measurement of Fluid Flow 7 Measurement of Level 8 Measurement of Stress-Strain 9 Measurement of Time Constant 10

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**Statistics Theory– Calibration**

Pressure Measurements using a Bourdon Gauge Statistics Theory– Calibration

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

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

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**Variable: Traits that can change values from case to case. Examples:**

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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**Variables may be: Independent or dependent In causal relationships:**

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. 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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**Nominal, ordinal, or interval-ratio **

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 The sum of the percentage column is 100%. 13

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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. 14

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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 Figure 2.2 An example of a frequency distribution histogram. The same set of quiz scores is presented in a frequency distribution table and in a histogram. Figure 2.3 An example of a frequency distribution histogram for grouped data. The same set of children’s heights is presented in a frequency distribution table and in a histogram.

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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 Figure 2.5 An example of a frequency distribution polygon. The same set of data is presented in a frequency distribution table and in a polygon. Figure 2.6 An example of a frequency distribution polygon for grouped data. The same set of data is presented in a grouped frequency distribution table and in a polygon.

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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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Figure 2.9 The population distribution of IQ scores: an example of a normal distribution.

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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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**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**

Figure 2.11 Examples of different shapes for distributions.

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**Como Construir un Histograma**

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

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**Calcular valor máx. y min. **

Pasos a paso Calcular valor máx. y min. Calcular la diferencia entre máx. y min. R= rango. Calcular el numero de clases h=1+3.32*log(n). Calcular ancho de clase C= R/h. Construya el histograma con los datos obtenidos. 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**

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**

En la celda B14 escribes la formula y le das Enter

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**Calcular los valores Máximo y Mínimo**

En la celda B15 escribes la formula y le das Enter

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**Calcular del Número de datos**

En la celda B16 escribes la formula y le das Enter

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Calcular el Rango 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**

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

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**Construir las Clases (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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**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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**It is simply the ordinary arithmetic average. **

Measuring the Mean Notation: 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 ?

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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

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Pressure Bourdon Gauge

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**In Fluid Mechanics we often use gage pressure and vacuum 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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**Pvac=-Pgage= Patm - Pabs**

Pressure Absolute Pressure Force per unit area exerted by a fluid Gage Pressure Pressure above atmospheric Pgage= Pabs - Patm Vacuum Pressure Pressure below atmospheric Pvac=-Pgage= Patm - Pabs

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**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 = 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: The pressure change across a fluid column of height h is ΔP=ρgh Pressure increases downward in a given fluid and decreases upward (i.e., Pbottom>Ptop), and 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 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. 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. Hysteresis is usually specified for a measurement system in terms of the maximum hysteresis error found in the calibration, ehmax, as a percentage of full-scale 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): 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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Application Problems

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

Homework1 WebPage Due, Wednesday, February 02, 2011 Omar E. Meza Castillo Ph.D.

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