Lucky Lindy

 

In the 20’s aerial records awaited to be shattered: first the North Pole in 1926 claimed by LCDR Richard E. Byrd, the first solo trans-Atlantic crossing in 1927 by Charles A. Lindbergh, and the first crossing of the South Pole by Byrd in 1928.  Lindbergh selected a single engine monoplane designed by Ryan Aircraft.  It featured a single Wright Whirlwind J-5C engine.  Both of Byrd’s aircraft were tri-motor capable of flying on two engines.  Lindbergh carefully chose his aircraft powered by the reliable Wright Whirlwind engine.  He kept the weight of his aircraft intentionally light—no sextant, parachute, or radio.  His flight of 33.5 hours covering 3,150 nmi was successful.

Assume that Lindbergh had the option to select a one-engine, two-engine or three-engine aircraft.  The three-engine aircraft he dismissed as unaffordable.  The two-engine aircraft of the era required that both engines function to sustain flight.  Lindbergh must decide which configuration is best suited to achieve success.  In either choice the engines are identical and come from the same production lot.  He decides:

a.     The single engine aircraft is 100 percent safer than the dual engine aircraft option

b.     The dual engine aircraft is the preferable choice since the likelihood of failure is diminished twofold

c.     The single engine aircraft is preferred as it would be lighter and less complex

d.     The dual engine aircraft has a higher probability of survival

The answer is:

In the dual engine option we place the aircraft at double the risk.  The second engine permits greater gross weight and more fuel capacity, but the aircraft cannot sustain flight without it.  The likelihood for the second engine being defective is the same as the first.  Consider that the likelihood for a defective engine in a production lot is 0.5 percent.  In a lot size of 200 there would therefore be 1 defective engine.  In a one engine aircraft there would be a 1/200 chance that the engine selected would be defective.  In a two engine aircraft there would be a 2/200 (1/100) chance that an engine would be defective.  In this comparison of aircraft engine configurations, Lindbergh has twice the likelihood of a defective engine being installed in the option of a two-engine aircraft.

Another viewpoint which arrives at the same conclusion is comparing the system failure and reliability of the two configur-ations.  Assume that the Whirlwind engine has a 200 hour (0.005 failures per hour) mean time to failure (MTTF).  With two engines (both must function to sustain flight) the total failure rate is the sum of the individual failure rates 0.005 x 2 failures per hour or 0.01 failures per hour (the two engines are treated as operating in series thus a single failure is a total failure).  The two-engine configured aircraft would experience, on the average, an engine failure once every 100 hours of flight that is twice the failure rate of the single engine configuration.  In the former both engines have to function to sustain flight.  The engine configur-ation reliability comparisons are contained in Table 4 in the first two columns.  The case of dual engines in parallel (assuming newer technology permits a single engine flight) is shown in the third column.  The last two columns portray three engines in parallel and three engines in parallel with at least two operational respectively.  Note that the MTTF (for a single engine failure) of the dual configuration (two engines must function option) is 100 hours versus the MTTF of the single engine configuration which is 200 hours.

Reliability is the probability that a unit will continue to operate up to time 't' in a steady state operation.  Reliability is defined as the negative exponential distribution and expressed as:

R = e-lt

where ‘l’ is the failure rate and ‘t’ the duration of the operation. 

The mean time to failure MTTF (hrs) for a unit is expressed as m = 1/l.  A parallel system with “n” units in parallel will fail only if all units fail giving rise to the expression, “redundancy to the nth degree.”  Redundancy enables higher system reliability to be achieved with units of lower reliability.

Lindbergh’s triumphal non-stop solo flight from New York to Paris in August 1927 in the Spirit of St. Louis was not only a first, but is frequently cited as an example of exceptional planning, pilotage, and dead reckoning.  Lindbergh drew a straight line on a gno-monic projection chart between New York and Paris, knowing that a straight line approximated a great circle on this chart.  He divided the flight path into 100-mile segments (approximating hour length legs) and transferred the coordinates at 100-mile intervals to his Mercator map (where a straight line is a rhumb line).  Short rhumb line segments preserve the conformance to an overall great circle course.  He marked each

Table 4.  Engine Configuration Reliability Comparisons

Engine Configuration

Single

Dual (Engines In Series)

Dual (Engines In Parallel)

Triple (Engines In Parallel)

Triple (With At Least
Two Engines Operational In Parallel)

Failure rate l 0.005 failures/hr/ engine

 

 

 

 

 

MTTF (hrs)
m =   one failure

200

 

 

 

 

ms =   one failure

 

100

 

 

 

mp =   two failures

 

 

300

 

 

mp =          three failures

 

 

 

367

 

mp =   one failure

 

 

 

 

167

Probability of Mission Success

 

 

 

 

 

Ps   = e-lt = e-0.005(33.5)

0.845776

 

 

 

 

Ps   = e-lt.e-lt = e-2lt

 

0.7153381

 

 

 

Ps   = 1 - (1 - e-lt))2
        
= 1 - (1 -  0.8457766)2

 

 

0.9762151

 

 

Ps   = 1 - (1 - e-lt)3
        
= 1 - (1 - 0.8457766)3

 

 

 

0.9963318

 

Ps   = e-2lt(3 - 2e-lt)
      = 0.7153381
         (3 - 2 x 0.8457766)

 

 

 

 

0.9359818

Symbol Key
l    failure rate per hour                               m   mean time to failure MTTF (hours)
t    mission time (hours)                              R    reliability

 

leg with its magnetic heading (based on the forecast winds).  The great circle course shaved 140 nmi from the flight over a rhumb line course (shown in Figure 11 ).

Figure 11.  Great-Circle and Mercator Course Comparisons to Paris from New York (Mercator Chart)

The resulting polygonic line curved gracefully through New England, Nova Scotia, and Newfoundland, eastward over the Atlantic, down past the southern tip of Ireland, across the narrow strip of England, and terminated over Paris.  His main concern was obtaining maximum range from his Ryan monoplane and he traded this off against additional navigational aids such as a radio and sextant.  He used a Pioneer Eclipse Earth inductor compass which utilized a rotating coil in the Earth’s magnetic field to determine magnetic heading.

He flew as low as 20 feet over the water, relying on observing the waves to determine the drift correction.  He was only six miles off course as he made landfall at Nova Scotia and projected this error to be less than 50 miles when he reached Ireland.  He was within 3 miles of the southern tip of Ireland upon his arrival and then continued his course to Paris, which he reached after covering 3,150 nmi in 33.5 hours.  Lindbergh was dismayed when reporters referred to him as “Lucky Lindy” in his triumphal navigational achievement.  He attributed his success to his meticulous planning and the Earth inductor compass.  In reality his amazingly precise landfall at Ireland, 3 miles off after a span of 1,700 miles over water, represented an incredible accuracy of 0.1 degree course adherence.  Considering that he was dodging adverse weather (which delayed his planned arrival by an hour) the inherent 1 to 2 degree compass error, the uncertainty in magnetic variation and rough measurements of drift by observing the waves, he should have been by his own earlier projection 50 miles off course.  He had no accurate way of measuring his groundspeed.  There was unquestionably a combination of compensatory errors and luck.  Lindbergh knew that he could not miss the European coast as revealed in his book, The Spirit of St. Louis.