INTRODUCING ERRORBy: Tala Rifka, Samantha Jones, Anthony Scheller,

Zach Palmer, Tyler Droog

Introduction and Methodology

● Both error and units are necessary, important components of engineering

● Unit conversions can cause serious issues, including in some extreme cases, death and bodily harm

● Error can also be a factor in calculations that can cause issues

● It is important to keep both of these factors in mind when solving engineering problems

● Our paper outlines a lot of the concepts of miscoverted units, error and the necessity of both uniformity among units and use of error terms in calculations.

http://www.alderkoten.com/ws/2012/05/engineering-recruiters-access-mexicos-talented-engineers/

Types of Error ● Conversion

● Rounding

● Truncation

● Human

● Uncertainty

http://blog.creativesafetysupply.com/wp-content/uploads/2013/10/Critical-Errors-What-to-be-Aware-of-and-What-to-Avoid.png

The NASA Disaster● The NASA Mars Climate Orbiter reached Mars on 23rd September 1999

to establish an orbit around Mars but it disappeared.

● The engineering team used Imperial measurements while the Jet Propulsion Lab Team used the metric system.

● As a result, the wrong navigation information was sent to the Orbiter and NASA lost $125 million.

http://mathspig.files.wordpress.com/2009/10/mars_climate_orbiter_during_tests.jpg

http://upload.wikimedia.org/wikipedia/commons/1/19/Mars_Climate_Orbiter_2.jpg

http://blogs.msdn.com/blogfiles/research/WindowsLiveWriter/WTFFandtheMarsClimateOrbiterMCOcrashin19_C08E/image_thumb_1.png

Mars Orbiter was doing a aerobraking maneuver to enter into Martian orbit, and when it started it’s rocket motor, it blasted itself out of orbit and crashed.

Patriot Missile Failure On February 25, 1991, during the Gulf War, an American Patriot Missile battery in Dharan, Saudi Arabia, failed to intercept an incoming Iraqi Scud missile. The Scud struck an American Army barracks and killed 28 soldiers

● The software truncated the 24-bit numbers

when converting to binary

● This truncation error grew the longer the

system ran

● At the time of the failure, the system had been

running for over 100 hours

http://ta.twi.tudelft.nl/users/vuik/wi211/patriot.jpg

Explosion of Ariane 5On June 4, 1996 an unmanned Ariane 5 rocket launched

by the European Space Agency exploded just forty seconds after lift-off

● The cause was a software error in the inertial reference system

● A 64-bit floating point number was meant to be converted to a 16-bit signed integer

● The number was larger than 32,678, which is the largest storable number for 16-bit, so the conversion failed

● The development of the rocket cost over $7 billion

● The rocket and its cargo cost $500 million

http://ta.twi.tudelft.nl/nw/users/vuik/information/ariane5.jpg

Air Canada runs out of fuel● In 1983, an Air Canada plane ran out of fuel in the middle of the flight.

● It was Air Canada’s first time to use metric units instead of imperial units but not everybody knew.

Flight 143 after landing at Gimli,Manitoba

http://upload.wikimedia.org/wikipedia/en/2/24/Gimli_glider.JPG

Medical Mishap● In 1999, a patient was given 0.5 grams of Phenobarbital (a sedative) instead

of 0.5 grains (1 grain is about 0.065 grams).

● The Institute for Safe Medication Practices then emphasized that only the metric system can be used for prescribing drugs.

http://www.clinicalpharmacology.com/apps/images/photo_us_h/038/phen020b.jpg

Heavy Landing● In 1994, an American International Airways flight landed 15 tons heavier

than it should have.

● The problem was in kilogram-to-pounds conversion.

http://www.stmary.ws/highschool/physics/home/notes/dynamics/gravity/poundKg.jpg

http://img.planespotters.net/photo/176000/original/N625PL-Kalitta-American-International-Airways-Boeing-747-100_PlanespottersNet_176641.jpg

Columbus’ Arrival● Columbus miscalculated the circumference of the earth as he used Roman

miles instead of nautical miles.

● This is why he arrived to the Bahamas on October 12, 1492 and thought he arrived to Asia.

http://mentalfloss.com/article/25845/quick-6-six-unit-conversion-disasters

http://www.operationworld.org/files/ow/maps/lgmap/baha-MMAP-md.png

Truncation● Truncation errors occur from using an approximation

instead of an exact mathematical procedure.

● Example:

derivative of velocity by a finite-difference equation:

dv/dt ~= [v(ti+1)-v(ti)] / (ti+1-ti)

http://www.sosmath.com/calculus/tayser/tayser01/tayser01.html

Vancouver Stock Exchange ● In 1982 the Vancouver Stock Exchange instituted a new index

initialized to a value of 1000.000. The index was updated after each transaction. Twenty two months later it had fallen to 520.

● The cause was that the updated value was truncated rather than rounded. The rounded calculation gave a value of 1098.892.

http://cdn-7.stocks-for-beginners.com/image-files/vancouver-stock-exchange.jpg

Round Off ● Roundoff errors occur because digital computers cannot represent

some quantities exactly. This rounding can lead to erroneous results.

● Examples:

● Large computations

● Floating numbers

● Arithmetic operations

http://blogs.mathworks.com/seth/2009/12/14/round-off-error/

The sinking of Sleipner A● The Sleipner A platform produces oil and gas in the North Sea at a depth of

82m

● The concrete structure that supported the platform sprang a leak and sank

● The total economic loss was estimated to be ~$700 million

http://ta.twi.tudelft.nl/users/vuik/wi211/sleipner.gif http://ta.twi.tudelft.nl/users/vuik/wi211/sleipnera.jpg

The sinking of Sleipner A ( continued)

● The post accident report found the error to an “inaccurate finite element approximation of the linear elastic model of the tricell (using the popular finite element program NASTRAN)” due to rounding errors

● The shear stress was underestimated by 47%

● The photo on the right shows the crack in the concrete support

http://ta.twi.tudelft.nl/users/vuik/wi211/tricell.gif

http://www.ima.umn.edu/~arnold/disasters/tricell-diag.gif http://ta.twi.tudelft.nl/users/vuik/wi211/sleipner-cells.gif

http://www.google.com/imgres?imgurl=&imgrefurl=http%3A%2F%2Fwww.globalspec.com%2Freference%2F13654%2F179909%2Fappendix-a-units-conversion&h=0&w=0&sz=1&tbnid=hK6YJYGu8olurM&tbnh=229&tbnw=220&zoom=1&docid=89kkkl7VQSnJtM&hl=en&ei=LXmSUrtL6MPZBYGTgNgF&ved=0CAMQsCU

Derived Units● Length, mass and time are the most basic

dimensions that allow for the composition of other units.

● All other units can be derived from these basic dimensions.

http://pleasediscuss.com/andimann/20091204/cio-dilemma-balancing-tactical-and-strategic-projects/875412_330130201/

http://www.branfordbuzz.com/2013/11/02/fallback-starts-sunday-set-those-clocks/

http://www.comicvine.com/forums/off-topic-5/ask-your-ruler-1481725/

Derivations● Derivations arise from need: for example

Newton’s law states that F = mass*acceleration.

○ Mass has units of kilograms (kg).

○ Acceleration has units of meters per second squared (m/s^2).

○ The resulting unit of Force is equivalent to a kilogram meter per second squared (kg*m/s^2).

● To help shorten the names of units, specific combinations of units can be called something else.

○ For example, 1 kilogram meter per second squared (kg*m/s^2) is equal to 1 Newton (N).

http://www.shutterstock.com/pic-72593407/stock-photo-red-apples-on-apple-tree-branch.html

Derived Units● There are 2 different unit systems English and SI (Systeme International).

● There are conversions needed to get from one system to the other

● There are issues associated with going between systems.

http:// chem105.tripod. com/id6.html

Direct and Functional Unit-Conversion Factors

● Issues between U.S. and U.K. units

○ Both use English system

○ There is a need to be specific when converting to units like gallons (gal)

■ There are imperial gallons(U.K. units) and regular gallons (gal) that the U.S. uses.

magnusbirkner .blogspot. com

Examples of conversions

http://3d-pictures.feedio.net/convert-measurement-units-using-unit-conversion-tool/techyard.net*wp-content*uploads*images*UnitConversionTool_DD54*units_of_measure.gif/

https://www.smore.com/8emn-customary-unit-conversions

Examples continued

math.tutorvista.com

SI Base Units

www. boundless. com

www. processassociates. com

Types of conversion factors● Two types of conversion factors:

1. Direct unit transformation factors

2. Functional unit transformations (or affine transformations)

http://www.copper.org/resources/properties/cryogenic/

Direct Unit Transformation FactorsUnit conversions usually require applying an equation, or the multiplication a series of conversion factors.

● An example is the conversion between the absolute temperatures Kelvin and Rankin

T(R)= 1.8*T(K)

● This is the most common way used in unit conversions

http://www.mercury.utah.gov/images/thermometer1.jpg

Functional Unit TransformationsThese require certain definitions such as some temperature unit conversions and the conversion between absolute and gage pressure.

● T(0 C)+ 273.15= T(K) T(0F)+ 459.67= T(R)

● P(psia)= P(psig)+ 14.7

www. 101qs. com

Functional unit transformations (continued)

● Some functional unit transformations are more complicated than just a simple addition

● An example is the conversion between the gas volume (ft3) and the gas volume at standard state(scf or standard ft3) which depends on the gas conditions and thus requires the use of the real gas equation of state

PV= ZnRT

where P is the absolute pressure, V is the gas volume, Z is the real gas deviation factor, n is the number of moles, R is the universal gas constant and T is the absolute temperature

http://www.bbc.co.uk/bitesize/higher/physics/images/pressure_1.jpg

Steps for converting units● Substitute the desired units.

● Multiply each variable by unit conversion factors to convert them to the prescribed units of the equation.

● Combine the various conversion factors into lumped factors.

● Round the answer off to the correct number of significant figures.

http://accessiblemediacenter.techadapt.com/samples/CAST_Exemplars/Exemplar6/content/images/U01C08/EQp146-004.jpg

http://blogs.mathworks.com/seth/2009/12/14/round-off-error/

Example: Mean hydraulic diameter● The mean hydraulic diameter D (m) of capillary flow paths in porous

materials in SI units can be found by

● where τ is the tortuosity (dimensionless), k is the permeability in m2,and ϕ is the porosity of the porous media. We want to convert this equation to units of μm2 for the mean hydraulic diameter, permeability in millidarcy and porosity as a percentage.

Mean Hydraulic Diameter example continued

● After inserting the proper unit-conversion factors in the equation on the previous slide, the following results:

● And by combining all the conversion factors:

● In order to check the converted equation, we solve it using the desired units and compare the resulting numerical value with the value obtained from the original equation.

Common Mistake● A common mistake everyone tends to make is to start with the original units

and convert them to the desired units while the opposite is actually correct.

● Relating to the example we considered in the previous slide, if this mistake was made we would get the following equation:

● Which would simplify to the following result which is incorrect.

Matlab Conversion Code● To demonstrate the capabilities of modern technology and

applications of Numerical Methods, we have created a simple program in Matlab designed to convert between Celsius, Kelvin, Rankine and Fahrenheit.

● The next couple of slides document the program and break it down.

Matlab Conversion CodeFigure created by Group 06Matlab code can be found in the notes below

Matlab Conversion Code

Figure created by Group 06

Matlab Conversion Code

Figure created by Group 06

Outputs & Testing

Figures created by Group 06

Output & Testing

Figures created by Group 06

Final Remarks● The NASA incident as well as the other examples above that

occurred as a result of units could and should have been prevented.

● Engineers must take unit conversions seriously and be very careful when handling units.

● Clear unit conversions shown and excellent communication are some of the best things that can be done to prevent unit errors.

● The most important thing an engineer can do to prevent unit errors is to understand the importance of units and the seriousness of the errors that can occur as a result.

● Both units and error are important factors to think about when doing calculations that have the possibility to hurt other people.

References/Resources● http://research.che.tamu.edu/groups/Seminario/numerical-topics/units

%20problems.pdf

● http://mentalfloss.com/article/25845/quick-6-six-unit-conversion-disasters

● http://www.cnn.com/TECH/space/9909/30/mars.metric.02/

● http://ta.twi.tudelft.nl/users/vuik/wi211/disasters.html