  # Measurements and Analysis for Mechanical Engineers

An online textbook geared toward mechanical engineers that teaches experimental design, data analysis, uncertainty concepts, and key measurement techniques.

# Introduction to Measurements

​After this chapter you should be able to:

• Identify the parts of a measurement system
• Explain the difference between independent, dependent, and extraneous variables
• Explain the difference between continuous and discrete variables
• Properly use and convert units
• Discuss the significance of standards for units

## How and Why we Measure

What is the temperature of the room you're in right now?

It seems like a simple question. Perhaps you have a thermostat on the wall which tells you what the temperature should be. But is that temperature the temperature of the room? Is it different near the window? If you used a standard liquid in glass thermometer, would the temperature be the same as the thermostat? If you used a thermocouple, would it be the same as your thermometer? If the room was large and you measured temperature in several places, would they all be the same?

Measuring engineering variables is rarely simple and straightforward. But if we understand how measurement systems work, what errors we are likely to see and how to correct for them, and what to do with the data when we get it, we can get answers to our engineering problems. Figure 1 shows a big picture map of the concepts we need to understand in this course. We will spend a lot of time explaining what measurements systems are commonly used in mechanical engineering and how they work. This is important - some measurement systems cannot be used in certain situations because of how they are designed. For example, if you wanted to measure strain in a contact lens, you couldn't glue a metallic strain gauge on it! We also need to understand what we do with measurement systems, which is measure engineering variables. To do this, we need to know which variables we are controlling and which we are measuring. We need to know if our answer is the correct answer. Is our device calibrated and reading correctly? Do we know how accurate our device is before we start? Do we know what might come up while we are measuring that could give us a wrong answer? How good do we need our answer to be? Finally, we need to do something with that answer. We need to analyze our results and present them in a way that others can understand and benefit from. Otherwise, we've done an experiment to no purpose.

## Measurement Systems

Before we start assembling equipment or choosing sensors, we need to consider why we are doing the measurement. Design of experiments will be covered in a separate chapter, but some of the key ideas should be understood early on. It is important to spend sufficient time developing the problem statement and the need for the experiment. I could do a week long series of experiments to determine the melting point of lead experimentally, using highly accurate calibrated temperature devices and a carefully controlled furnace. However, I could just as easily look that up in nearly any materials science textbook, or online. No matter how carefully planned and executed the experiment is, it wouldn't provide us with any new information. But if I am trying to determine which brand of soda has the lowest pH, this information might not be readily available for all types of soda. An experiment could solve this problem. I do still have to ask the question: What do I do with this information? If I determine the melting point of a metal, I can use that information to determine if that metal can be used in a particular situation. If I am trying to avoid tooth decay, I may want to know the pH of the sodas I tend to drink. Engineers have a large number of things they might want to measure, but in all cases, there has to be a reason for it.

Once it has been established that a measurement of some sort is truly needed, a measurement system must  be chosen.  An experiment or process will generate some sort of effect that we want to measure - an increase in temperature, a decrease in pressure, or a change in concentration of a chemical species. A measurement system is any instrument or series of instruments used to detect physical variables and turn them into an output that we can use. Figure 2 below shows a diagram of a generic measurement system. The arrows indicate the flow of a signal through the system. The input signal can be an actual electrical signal, but it can also be temperature, pressure, strain, or any other variable we are trying to detect. The detector or sensor detects the variable which we're trying to measure. We may want to compare our signal to a calibration source, which gives us a known value to compare our measured value to. This will tell us if our sensor is reading accurately or not. The sensor is generally coupled with a transducer which turns the measured variable into something else, such as a movement on a dial or an electrical signal. Sometimes the transduced signal needs to go through a signal conditioning stage where the signal can be filtered, amplified, or otherwise modified. The signal then has to be connected to some sort of output, such as an indication on a scale, a digital read out, or a data acquisition system. Finally, in some cases the output signal is used for a feedback or control stage, in order to control another system, such as a heating system.

Figure 3 below shows an example of an extremely simple measurement system - a liquid in glass thermometer. This measurement system has only two stages. The liquid in the bulb is the sensor-transducer stage, which reacts to changes in temperature by expanding or contracting. It turns a temperature change into a movement of the liquid. The output stage is the scale on the side of the thermometer - it indicates the temperature. A device this simple does not require an intermediate stage or a feedback stage.

Identifying the stages of a measurement system
No correct answers: No correct answer has been set for this question

Consider a standard room thermostat. Determine the sensor, transducer, output, and control stages for this measurement system.

## Variables

The purpose of a measurement system is to detect variables. In any experiment, we are trying to measure the effect of changing one thing on something else. Perhaps we have a closed vessel of a known volume, filled with air. If we heat this vessel, the pressure will increase, due to the ideal gas law:

where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature. In this example, n, V, and R are not changing, so they are not variables in this context. If we are systematically varying temperature, then we call temperature the independent variable. The independent variable is what we are systematically controlling or changing. The variable that is in control is being held at a constant value during an experiment. Complete control of a variable is usually not possible, but generally one can get close enough to see the effect of the independent variable on the dependent variable.  The dependent variable changes when we change the independent variable - it is the effect we're trying to measure. In this example, pressure is our dependent variable, since it changes as we change the temperature. In addition to these variables, we may also have extraneous variables. Extraneous variables are variables we're not able to control, but which might affect the outcome of the experiment. For example, if our pressure measurement device is referenced to atmospheric pressure, any change in atmospheric pressure might change our pressure reading. We can't control the atmospheric pressure, so it is an extraneous variable. Other possible extraneous variables might be electrical noise sources from nearby equipment, which could affect the reading on an electronic transducer.

A more concrete example of the effect of extraneous variables can be seen in Figure 4. The data represents a series of experiments to measure the boiling point of water under controlled conditions on three different days. Here, the boiling point of water appears to change due to the error caused by atmospheric pressure changes from day to day. If the water is pure and unadulterated, and the experimental conditions are the same for each experiment, the boiling point of water should be the same every time. The fact that it is not points to the effect of the changing atmospheric pressure from day to day. Sometimes the extraneous variables in an experiment are not easy to recognize. This is why careful analysis of the process we are measuring as well as the equipment we are measuring with is crucial to making sure that our answers are meaningful. Figure 4: Data from an experiment to measure the boiling point of water, showing the effect of extraneous variables.​

​Experiments may have multiple dependent and independent variables. Sometimes those variables may be the same physical quantity being measured. For example, let's consider a case where I have a gas regulator for an air line, and I want to know how accurate it is. I could set that regulator to 60 psi, and then connect the air line leaving the regulator to a calibrated gauge. In this case, the independent variable would be pressure - the pressure coming out of the regulator. The dependent variable would also be pressure - the pressure being measured by the calibrated gauge. In fact, an extraneous variable would also be pressure - atmospheric pressure. It is therefore important to consider in every experiment: what is being systematically varied, what is being measured, and what can change our outcome that we cannot control.

Identifying variables

A student is doing an experiment to determine which common beverages have the lowest pH. The student chooses 5 beverages: cola, a sports drink, seltzer, orange juice, and coffee. The pH meter works best at room temperature, but the student chooses to do the test outside on a 95 degree day. Identify the dependent, independent, and extraneous variables in this experiment.

Premise
Response
1

Type of beverage

A

Extraneous variable

2

pH

B

Independent variable

3

Temperature of surroundings

C

Dependent variable

​The variables that we measure can be continuous -- meaning that the variables are continuously changing, even if they approximate a mean -- or they can be discrete -- producing a single number. The temperature in an oven that is heating up would be an example of a continuous variable. The number of students in a classroom would be a discrete variable - there can be 5 students in a room, or 6, but never 5.75 students. Discrete variables can also be on an ordered scale, meaning the value is numerical, or a disordered scale, meaning the value is not numerical. An example of a disordered scale would be something like the type of shoes people are wearing while running the Boston Marathon.

Types of variables

Match each variable to its type

Premise
Response
1

The number of dogs of each breed at a dog show

A

Discrete on a disordered scale

2

The breed of dogs that walk by a particular bench at a park in an hour

B

Discrete on an ordered scale

3

The heart rate of a dog during exercise

C

Continuous

The type of variable is important when we consider how we are going to record that variable. Continuous variables may need to be recorded with a computer based data acquisition system. We could also decide to sample continuous variables at regular time intervals. This is not uncommon - we just need to be aware that we might be missing data that occurs between our sampling points. We are in effect turning our continuous variable into a discrete series of variables by sampling at discrete time intervals. Disordered variables need to be recorded by hand, or with a survey instrument. If I want to know how many fans of a given sports team are in my class, I can't hook students up to a data acquisition system, but I could have them fill out an online survey. When we design experiments, the methods required to gather the different variables need to be thoroughly considered before we start.

## Units and Standards

Variables need to have units specified in order to make sense. Units need to be based on some physical reality. For example, force is given in units of Newtons (N). In SI units, Newtons are defined as:

Force is defined as mass x acceleration. Since mass is given in kg, and acceleration is given in m/s2, we can see that the units, like the variables themselves, make sense with physical reality. Primary units define the standard quantitative value for a particular variable. Derived variables are a combination of primary variables. Some of the common primary and derived units are listed in the table below. Although SI units are more common internationally and are used more commonly in engineering research, it is helpful for engineers working in the U.S. to also be aware of the Imperial units (IP) still in widespread use.

 Unit SI IP Primary Length meter (m) inch (in) Mass kilogram (kg) pound-mass (lbm) Time second (s) second (s) Temperature Celsius (C) or Kelvin (K) Fahrenheit (F) Current ampere (A) ampere (A) Derived Force newton (N) N = kg/m.s2 pound-force (lbf) Voltage volt (V) V = kg·m2/s3·A1 volt (V) Resistance ohm (​Ω) Ω =  ‎kg⋅m2/s3⋅A2 ohm (​Ω) Stress pascal (Pa) Pa = ‎kg /m⋅s2 pound-force/inch2 (psi) Energy joule (J) J = ‎kg⋅m2/s2 British thermal unit (BTU) Power watt (W)  W = ‎kg⋅m2/s2 foot pound-force (ft-lb)

It is a good habit to get into to always check your units for consistency. If your units don't cancel out to what you expect in the end - that's a clue that there's a mistake somewhere.  It is especially important to note the units for any constants that you look up, either in a textbook or online. If one source provides information in kilojoules, and another in joules, your answer could be off by 1000 if you are not aware. There are a number of excellent unit conversion calculators online. A particularly useful calculator can be found at https://www.digitaldutch.com/unitconverter/speed.htm

Units conversion

You find a pressure gauge in the lab and the spec sheet says that the maximum pressure it can read a maximum of 2 inches of water. What is the maximum pressure in pascals?

Units are based on some standard. Typically this is some reproducible physical phenomenon, like the distance that light travels in a certain period of time, or the number of decay events of a radioactive element. These have been defined over the years by metrologists - scientists who study measurements. Organizations such as the National Institute of Standards and Technology (NIST) and the International Organization for Standardization (ISO) are responsible for setting and maintaining standards for units as well as coming up with regulations and procedures for testing and measuring. These organizations and others are a good source of information on the best way to measure things properly. Standards may need to be purchased - such as using a calibration weight to determine the accuracy of a scale. Standards may need to be generated in the lab - such as when we use an ice bath to calibrate a temperature device. More information on standards and calibration will be discussed in subsequent chapters.

Metrology is still an active field of study - scientists are still coming up with better ways to measure things and standardize those measurements. The video below describes how the kilogram, currently based on a standard metal object in France, is being converted to a measurement based on Planck's constant.

In future chapters, we will be discussing accuracy, calibration, and other concepts that allow us to answer the question: What's the 'real' answer? How well do we know that answer? Is the temperature of this room 75° F +/- 4°F? Or is it 75° F +/- 0.1°F? This type of question will be explored as we discuss instrument accuracy, precision, and uncertainty. We will also discuss how to look for trends in the data by graphing it, looking for correlations, and performing statistical analysis. This provides us with meaning for our data. If the temperature of the room varies by ten degrees over the course of the day, every day, maybe it is time to adjust the heating and ventilation system! Properly analyzed data allows us to make informed decisions and justify our conclusions.

## Image Sources

• Figure 3 created by Tristan T. R. Smyser
• All other images created by author.