Marginal Design

Marginal Design

An aspect of the cost of design is illustrated in the graph below, it is a general illustration that shows that as you design for increased completeness the costs rise exponentially. Additionally, because the curve is asymptotic at completeness equals one hundred percent, total completeness is impossible. An example of this curve is removing a pollutant from water. To remove the first fifty percent of the pollutant costs about twelve dollars, but to remove ninety percent of the pollutant costs about ninety dollars. To remove one hundred percent of the pollutant would cost infinite dollars, it can’t be done.

The job of the engineer is to decide where along this curve her design is to lay. This marginal design requires the engineer to balance the cost of a design with the public’s willingness for failure of the design.

Another aspect of marginal design can be illustrated by the graph below that shows as you reduce the consumption of time or other resources. For instance, assume that a country is reducing its consumption of oil where one hundred percent of consumption is happening now. In order to produce or procure that last five percent of oil takes a large amount of “Difficulty” where difficulty is the dirtiest, most expensive, or other undesirable characteristic. Looking at the situation another way, by reducing oil consumption from one hundred to ninety five percent of the current total, the dirtiest and most expensive oil is forgone. In other words, reducing the first five percent of oil is much more valuable than reducing consumption from, for instance, fifty to forty five percent.  

The combination of these two curves demonstrate the power of designing at the margin, a significance that is often lost on the general public.

Units

Units

Units are the alphabet of equations, but they are more useful than the ABCs because they allow you to understand how different units are related.

The base, or fundamental, units upon which all other units are derived are: mass, length, time, luminous intensity, electric current, temperature, and chemical amount of a substance. For this discussion we will discuss only those units derived from mass, length, and time which will be written as [M], [L], and [T]. Because the derived unit of force is so commonly used, it is often written as a base unit: [F].

As an example, Newton’s second law says that force equals mass times acceleration. Written in fundamental units it would read:

Where acceleration is length divided by time squared. Note the use of the negative exponent to indicate that time squared is in the denominator.

In the International System of Units (SI) the mass is expressed in kilograms, length in meters, and time in seconds. This results in Newton (N) as the force unit. In the Imperial system mass is expressed in slugs, length in feet, and time in seconds. this results in pounds-force (lbf) as the force units.

Newton’s second law is also used to calculate weight where acceleration (a) is the acceleration due to gravity (g). On the surface of the earth:

Calculating forces in Newtons is widely understood and creates little confusion, while calculating force in the Imperial system is often misunderstood. The key is to recall that the fundamental unit of mass is the little discussed slug. What is confusing is that the Imperial system also uses a pound-mass (lbm). A pound-mass is used for convenience and is not a fundamental unit of mass. It is simply defined as being numerically equal to pound-force. For example, an engineer’s body weight (recall that weight is a force) is 120 lbf. Because a lbm is numerically the same as a pound force, her mass is 120 lbm. But using Newton’s second law and given her weight and the acceleration due to gravity (g) of 32.2 feet per second squared, her mass in slugs is 3.73 (120/32.2).

The following table gives examples of derived units and their respective base, SI, and imperial units. As mentioned above, while force [F] is not a true base unit, it is shown as such because it is so commonly used.

Derived Unit	Base Units	SI Units	Imperial Units	SI Name	Imperial Name
Force				Newton (N)	pound-force (lbf)
Density				None	None
Specific Weight				None	None
Energy, Work, and Heat				Joule (J)	British thermal unit (BTU)
Power				watt (W)	horsepower (hp)
Pressure				Pascal (Pa)	Pounds per square inch (psi)

One advantage of working and thinking in base units is it can tell you what math is required to determine a specific result. For instance:

Given: A small wind turbine has a capacity of 3.00 kW and runs at full capacity for one hour

Find: The amount of energy produced.

Solution: Referring to the table above you can see that a kW is a unit of power, a Joule is a unit of energy. You also note that power has time cubed in the denominator and energy time squared, So, multiplying power times time equals energy. In this example the unit of time is seconds for both power and energy, so we will 3,600 seconds rather than one hour in the calculation.

In the following unit conversion examples we first start with a table showing unit equality. These tables can be found readily online. For these examples, we need only the four equalities shown. Note that multiplying a value by one minute divided sixty seconds is equivalent to multiplying by one because the numerator and denominator are equal.

Before converting ensure that you are converting from and to the same base units. In this first case, both gallons per minute and cubic feet per second are flow rates (i.e. volume per time) and can be converted. Start  with 175 gallons per minute, and, keeping track of units (by cancelling units), multiple by a series of “ones”.  The second example is solved in a similar manner.

Digits

A common error committed by students is to display too many digits in a numerical answer. Many engineers consider an excessive number of digits to be an incorrect solution The rules displaying digits can be summarized below:

1. The number of digits displayed indicates the precision of the measurements and other values used to arrive at the answer.
2. When adding or subtracting, the digits displayed are as precise as the least precise value being added or subtracted.
3. When multiplying or dividing, the answer is only as precise as the least precise value being multiplied or divided.
4. When in doubt, be reasonable.

When using scientific notation there is no ambiguity in the number of significant digits. For instance, has four significant figures. Likewise, also has four. Written without scientific notation 500.2 would be known to have four significant figures, but the number of digits in in 7,000 would be unknown. Depending on the type of work, it is common to not use scientific notation in engineering. The rules for determining the number of significant figures state that all nonzero digits are significant, all zeros between nonzero digits are significant,  and all digits written when using scientific notation are significant.

Examples of calculations using significant figures:

because the least precise number is to the nearest tenth resulting in an answer to the same precision.

because the least number of significant digits is one, so the answer is given to one digit.

When displaying a solution, the number of digits represents the precision of the solution. There are two basic kinds of numbers in calculations.  The first is a measurement, and the second is a count.  Measurements have a finite number of digits as a result of the instrument used to take the measurement and the human error in measuring the measurement. The number of significant digits is shown when the measurement is written. A count has an infinite number of digits, but the number of digits shown is a whole number with no digits in the decimal places. It is the units that identify counts. Counts are either a thing or a person. The example below demonstrates this.

  In this example each widget (a thing) weighs 2.4 N, so two widgets weigh 5.8 N. It is the widget weight (a measurement) that controls the number of digits.

Another way to think of the measurements is with the terms “accuracy” and “precision” which are shown below using the metaphor of a dartboard.

Measurements of a single variable such as weight or length are precise when they are close together and this is shown by a relatively large number of significant digits. Accuracy is determined by the closeness of the measurement to its “true” value. True is in quotations because of the fact that the true value of any measurement is always unknown and unknowable. One way to convince yourself that the true value of a measurement is unknowable is to imagine measuring the width of a room. Start by measuring with an increasing level of precision as shown in the following table.

Method	Value
Pace	33 ft
Yardstick	33.2 ft
Tape	33.18 ft
Laser	33.184 ft
More precise laser	33.1837 ft

As the table demonstrates, the width of the room can be measured to an ever increasing precision leaving its true value unknown