2.2 Introductory Mathematical Concepts
2.2.2 Measurement and Numerical Uncertainty
2.2.2.3 Significant Figures
As a final topic concerning uncertainty, we make brief mention about significant figures of a number. This topic applies to any numbers that we deal with, whether they are measured in a test or are being manipulated in a post-test data reduction or analysis. This fundamental topic is often covered early in one’s scientific or engineering education, but it is then often “forgotten”, especially with the use of calculators and computers that can spit out an unlimited number of digits.
The numerical uncertainty is indicated by how many meaningful digits, or significant figures, there are in a value. The order of magnitude of the last digit in the significant figures is the uncer-tainty. For instance, if we measure the airspeed of an airplane as 243.7 miles per hour, we have four significant figures, where we are certain of the first three digits and the fourth digit is uncertain.
Therefore, the uncertainty in our airspeed value is on the order of 0.1 mile per hour.
We must also be careful to maintain the proper number of significant figures in our calculations.
When numbers are multiplied or divided, the result should have the same number of significant figure as the number with the fewest number of significant figures in the calculation. So, if we multiply an airspeed of 243.7 miles per hour by a time of 1.45 hours, the result for distance should be 353 miles (even though our calculator can supply a result of 353.365 miles).
When numbers are added or subtracted, the result should have the same uncertainty as the highest uncertainty in the numbers being added or subtracted. So, if we add an airspeed of 10 miles per hour to an airspeed of 243.7 miles per hour, the resulting sum should be 253 miles per hour, since our value of 10 miles per hour has the greatest uncertainty, on the order of 1 mile per hour. We have
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been careful not to write the result as 253.0 miles per hour, as this would imply an uncertainty on the order of 0.1 mile per hour, which would be incorrect.
Fractions and integers are considered to have an infinite number of significant figures. For instance, in the equation y = 1
2x2, the fraction is exactly equal to one divided by two, with an infinite number of significant figures (0.50000 … ) and the exponent of x is exactly equal to 2, with an infinite number of significant figures (2.0000 … ).
Always remember that despite the infinite number of digits that are available to us with calcu-lators and computers, we should not infer a greater certainty in our numbers than is really there.
This helps us to avoid drawing incorrect conclusions about test data or numerical analyses.
Example 2.3 Tomahawk Cruise Missile Drag Uncertainty Analysis The present example, from [5], illustrates the use of an uncertainty analysis in the assessment of the aerodynamic drag of a flight vehicle. The lift and drag are two of the most important aerodynamic parameters that determine the performance and flying qualities of a flight vehicle. Since it is not possible to directly measure the lift and drag in flight, other basic parameters must be measured to calculate these forces. The propagation of the errors and uncertainties in these basic measurements results in an uncertainty in the lift and drag. This example shows the results of an uncertainty analysis, applied to the calculation of the total vehicle drag obtained from several flight test measured parameters.
The uncertainty analysis was applied to the AGM-109 Tomahawk air-launched cruise missile (ALCM), designed and built by the General Dynamics Corporation in the late 1970s (Figure 2.7).
The AGM-109 missile had an 18.25 ft (5.563 m) long, cylindrical fuselage, with a circular cross-section, and a gross weight of 2553 lb (1158 kg). The missile had a cruciform tail (four tail fins in a cross pattern) and a small, straight wing (zero wing sweep) with an area of 12 ft2(1.1 m2).
Propulsion was provided by a Williams F107 turbofan jet engine. After being launched from a military aircraft, the AGM-109 unfolded its wing and flew to its target at subsonic speeds using its turbofan jet engine. Flight testing of the AGM-109 was conducted as part of a US military
“fly-off” competition among various cruise missile designs in the late 1970s.
Figure 2.7 Raytheon BGM-109 Tomahawk cruise missile, similar to the AGM-109 cruise missile, proposed in the 1970s. The AGM-109 did not win the “fly-off” competition and was never produced. (Source: US Navy.)
k k Table 2.3 Effect of a 1% change in the independent, measured
parameters on the AGM-109 Tomahawk drag coefficient.
Independent, measured parameter Change in drag coefficient
Engine airflow calibration 1.2%
Engine core speed 10.9%
Engine fan speed 1.3%
Engine thrust calibration 4.0%
Indicated air temperature 0.8%
Inlet total pressure calibration 3.0%
Nozzle area 2.5%
Sea level temperature 4.0%
Static pressure 1.0%
Wing area 1.0%
(Source: Data from [5].)
The in-flight drag of the AGM-109 cruise missile could not be measured directly. Instead, other basic parameters (termed the independent parameters) were measured in flight, and the drag (termed the dependent parameter) was calculated from these independent parameters. The inde-pendent measurement parameters included those that defined the flight condition (air temperature and pressure conditions), geometry of the vehicle (wing area), and engine performance (engine airflow, engine fan speed, nozzle area, and core speed). Thus, the dependent parameter (drag) could be mathematically expressed as a function of these independent parameters. The uncertainty analysis used a numerical technique that perturbed the independent variables, in a functional expression for the drag, to estimate the uncertainty in the drag.
Table 2.3 provides selected results from the uncertainty analysis, where the effects of changes in the independent measurement parameters on the dependent variable (the drag coefficient2) are shown for a subsonic flight condition. The independent parameters in the uncertainty analysis included the in-flight measured parameters and the parameters associated with certain instrumentation calibrations. In calculating the drag coefficient, each independent measurement parameter was changed by 1%, while keeping all of the other independent parameters constant.
The table shows the resulting percentage change in the drag coefficient due to a 1% change in each independent measurement parameter. For instance, a 1% change in the measurement of the sea level temperature resulted in a 4.0% change in the drag coefficient.
The results of the uncertainty analysis provided several important insights about calculating the in-flight drag for the vehicle. It identified which measurements were the most critical, and the major sources of error, in calculating the drag. Based on Table 2.3, the drag calculation is most sensi-tive to the measurement of the jet engine core speed.3 If the core speed measurement is in error by just one percent, then the resulting error in the drag computation is about 11%. This under-standing of the measurement sensitivities helps determine where to invest time, effort, and funding in making test measurements, for example which measurements should be made more carefully or require higher accuracy sensors. Conversely, the uncertainty analysis also provides insight into
2The non-dimensional drag coefficient, CD, is defined as the drag, D, divided by the product of the dynamic pressure, q, and a reference area, S, usually the wing area. The drag coefficient and other aerodynamic coefficients are discussed in Chapter 3.
3The engine core speed is the rotational speed of the turbomachinery in the central core of the engine. The faster the core speed, the more air is being sucked into the engine and the higher the thrust. This is discussed in Chapter 4.
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which measurements are not as critical, hence, requiring less attention. By including the instru-ment calibrations, the uncertainty analysis indicates which calibrations are most critical. Table 2.3 indicates that the calibrations associated with the engine thrust measurement and the inlet total pressure are more critical for the drag computation. Thus, it would be worthwhile to expend more effort in performing these calibrations to obtain a more accurate value of the drag.