Uncertainty Analysis and Propagation of Uncertainty in Measurements, Lecture notes of Aeronautical Engineering

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Experimental Methods for

Engineers

Basic Concepts (Chapter 2)

By

Mohsen Soori / Mehdi Shahedi Asl

Basic Concepts: Definitions

• Readability (R): indicates the closeness with which the scale of the instrument may be read. an instrument with
• a 12-in scale would have a higher readability than an instrument with a 6-in scale and
• the same range. Readability < Readability
• Least count (LC): indicates the smallest difference between two indications that can be detected on the instrument scale. 37 º 37.5º

37.8 (^) LC = 0.

Definitions-continue

• Hysteresis Error (HE): related to the difference in readings depending whether the value of the measured quantity is approached from above or below.
• H.E. may the result of: friction, magnetic field, elastic deformation, thermal effects,…

Dimensions and units

• All physical quantities have a fundamental dimension that is independent of the units of measurement.
• Fundamental dimensions:
• L = length
• M = mass
• F = Force
•  = time
• T = temperature
• All other physical quantities used may be expressed in terms of these fundamental dimensions.
• Force N (Newton = kg× m/s 2 )

The generalized measurement system

• Most measurement systems may be divided into three parts:

1. A detector-transducer stage: detects the physical variable and perform either a mechanical or electrical transformation to convert the signal into a more usable form.
2. Intermediate stage: modifies the direct signal by amplification, filtering, or other means so that desirable output is available.
3. Final or terminating stage: indicate, record, or control the measured variable.

Generalized Measurement

System

The generalized measurement system;

Burdon-tube pressure gage

Detector- transducer stage Indicator stage Modifying stage Gearing mechanism

Definitions-continue

• Accuracy (A): “the closeness of the agreement between the result of a measurement and a true value of the measured.” It indicates the deviation of a reading from a known point.
• Usually expressed as a percentage of a full scale.
• Precision (P): indicates the ability of an instrument to reproduce a certain reading with a given accuracy.

Accuracy: Refers to how close a measurement is to the true or accepted value. Relatively low determinate error. Close to a ‘true’ value. Accurate and precise

Fundamental Methods of

Measurements

◼ There are two basic methods of measurement: ◼ Direct comparison: with a primary or secondary standard ◼ Indirect comparison: conversion of measurand input into an analogous form which can be processed and presented as known function of input ◼ - A transducer is required to convert the ◼ measureand into another form

◼ A procedure is said to be reliable if it may be completed with a high degree of accuracy and precision.

Definitions-continue

• Error: the deviation of an instrument reading from a known value.
• Calibration: a comparison of a particular instrument with another higher accuracy instrument, or with a known input source.
• Uncertainty: “a parameter, associated with the result of a measurement, that characterizes the dispersion of values that could reasonably be attributed to the measured.” Or: the range of instrument readings where the known value is not available.

Errors in measurements are

due to:

◼ 1 ) Systematic Errors:

◼ Procedural errors made by the experimenter.

They cause the measured value to deviate

from the "accepted" value in the same

direction i.e. always higher or always lower.

Examples of causes include miscalibration of

instruments and parallax errors. These errors

can be detected and corrected. Systematic

errors are calculated as Percent Error.

Some examples: Mis-calibrated instrument unaccounted cable loss Systematic Errors – cause the measured result to deviate by a fixed amount in one direction from the correct value. The distribution of multiple measurements with systematic error contributions will be centered some fixed value away from the correct value.