Category Archives: uncertainty

Wind power

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Winds are generated by uneven heating of the earth’s atmosphere by the sun, which causes hotter, less dense air to rise and more dense, colder air to be pulled into replace it.  Of course, land masses, water evaporation over oceans, and the rotation of the earth amongst other things added to the complexity of weather systems.  However, essentially weather systems are driven by natural convection, a form of heat or energy transfer, as I hinted in my recent post entitled ‘On the beach’ [24th July, 2013].

If you are thinking of building a wind turbine to extract some of the energy present in the wind, then you would be well-advised to conduct some surveys of the site to assess the potential power output.  The power output of a wind turbine [P] can be defined as a half of the product of the air density [d] multiplied by the area swept by the blades [A] multiplied by the cube of the velocity [v].  So the wind velocity dominates this relationship [P = ½dAv3] and it is important that a site survey assesses the wind velocity.  But the wind velocity is constantly changing so how can this be done meaningfully?

Engineers might tackle this problem by measuring the wind speed for ten minute intervals, or some other relatively short time period, and calculating the average speed for the period.  This process would be repeated over a long period of time, perhaps weeks or months and the results plotted as frequency distribution, i.e. the results would be assigned to ‘bins’ labelled for instance 0.0 to 1.9 m/s, 2.0 to 3.9 m/s, 4.0 to 5.9 m/s etc and then the number of results in each bin plotted to create a bar chart.  The number of results in a bin divided by the total number of results provides the probability that a measurement taken at any random moment would yield a wind speed that would be assigned to that bin.  Consequently, the mathematical function used to describe such a bar chart is called a probability density function.  Now returning to the original relationship, P = ½dAv3 and using the probability density function instead of the wind velocity yields a power density function that can be used to predict the annual output of the turbine taking account of the constantly changing wind velocity.

If you struggled with my very short explanation of probability density functions, then you might try the Khan Academy video on the topic found on Youtube at http://www.youtube.com/watch?v=Fvi9A_tEmXQ

Engineers use probability density functions to process information about lots of random or stochastic events such as forces ocean waves interacting with ships and oil-rigs, flutter in aircraft wings, the forces experienced by a car as its wheels bounce along a road or the motion of an artificial heart valve.  These are all activities for which the underlying mechanics are understood but there is an element of randomness in their behaviour, with respect to time, that means we cannot predict precisely what will be happening at an instant in time; and yet engineers are expected to achieve reliable performance in designs which will encounter stochastic events.  Frequency distributions and probability density functions are one popular approach used by engineers.  Traditionally engineers have studied applied mathematics that was equated to mechanics in high school but increasing they need to understand statistics.

Risk definition

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A section from a photoelastic model of turbine disc with a single blade viewed in polarised light to reveal the stress distribution.

Risk is defined as the possibility of something happening multiplied by the consequences when it does happen.  The public understanding of risk sometimes only extends to the first half of this definition.  Engineers seek to reduce the risks associated with component failure.  This means accepting a non-zero probability of failure happening and then designing for least catastrophic consequences.  So for instance in a jet engine, this implies designing so that if a crack develops it is in a blade rather than the disc to which all of the blades are attached.  The engine casing can be designed to contain a single blade breaking off and thus protect the rest of the plane from flying debris, but not to contain the rupture of an entire disc and set of blades.

For more information on the photoelastic stress analysis techniques used to generate the image, see http://www.experimentalstress.com

Unlikely failure

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High magnification image of a crack in polycarbonate viewed in polarised light which reveals fringes that a proportional to the stress in the material.

I used the term ‘unlikely to fail’ in my last post, in the context of engineering designs.  This might appear alarming, since people might assume that engineers design things to never fail.  However it is impossible to design with a certainty that failure will not happen.  There are several reasons for this, including: the conflicting requirements of less material to reduce cost and achieve sustainability and of more material to protect against failure; our lack of knowledge about in-service and exceptional operating conditions; and the extent, or otherwise, that the computational model used in the design analysis represents the real world.

Hence the phrase ‘unlikely to fail’.  We can reduce the likelihood, or probability, of failure, usually at additional financial and resource cost, but we can never reduce the probability to zero, i.e. there is always a risk of failure, although we do our best to ensure that designs are ‘unlikely to fail’.

Model validation

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Front cover of ASME V&V 10-2006, Guide for verification and validation in computational solid mechanics, American Society of Mechanical Engineers, New York, 2006.

Why is validation important?  Validation of computational mechanics models is defined as ‘determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model’, according to  ASME V&V 10-2006.  So, the validation of models of structural integrity for engineering design provides information about the degree to which the simulation results from the model can be believed.  This in turn helps in making decisions about how little material, and in what configuration, should be used to create elegant, sustainable designs that are unlikely to fail. So validation of computational mechanics models is an essential step in solving the ‘two earths’ dilemma (see post on August 13th, 2012).