Industrial Research 



279 



design of apparatus and in its subsequent manufacture 

 as well. 



Mr. Alger describes in general terms one situation 

 frequently met in research activities as follows: 



The first type of problem is one in wliich there are so many 

 different independent dimensions of a proposed shape to be 

 chosen, or in general so many independent variables, that it is 

 hopeless to find the optimum proportions by experiment. The 

 truth of this can readily be seen when it is realized that the num- 

 ber of test observations to be made increases exponentially with 

 the number of variables. If 10 points are required to establish 

 a performance curve for one variable, 1,000 observations will be 

 required if there are 3 independent variables, and a million if 

 there are 6 variables. 



As an illustration he cites the following problem: 



(g) An example of this kind of problem is that of designing a T 

 dovetail to hold the salient poles in place on a high speed syn- 

 chronous generator. A large machine of this type may have 10 

 or more laminated poles carrying heavy copper field coils, each 

 assembled pole weighing several tons and traveling at a surface 

 peripheral speed of 3 miles a minute. The centrifugal force on 

 each pound of the pole then amounts to approximately 500 

 pounds. The problem of designing dovetails to hold these poles 

 in place, even at over speed, is, therefore, one of great importance 

 and technical difficulty. For each such dovetail, there are 7 

 different dimensions which may be independently chosen. 

 While empirical methods have enabled satisfactory results to be 

 obtained in some cases, application of mathematics has recently 

 enabled marked improvements in dovetail designs to be made. 

 Generally speaking, these improvements have permitted an 

 overall strength increase of 20 percent to be obtained under 

 steady stresses and much higher gains to be made under fatigue 

 stress conditions; while at the same time the certainty of obtain- 

 ing the desired results on new designs has been very greatly 

 enhanced. 



A second e.^ample was brought to my attention by 

 Mr. L. W. Wallace, Director of the Engineering and 

 Research Division of the Crane Company: 



(A) A pipe fitting weighing several hundred pounds and in- 

 tended for high pressure service had a neck of elliptical cross- 

 section. As originally designed, the thickness of the casting 

 was intentionally not uniform, the variations having been intro- 

 duced empirically to strengthen it where strength was supposed 

 to be most needed. A redesign carried out on the basis of the 

 theory of elasticity showed the distribution of metal to be in- 

 efficient and resulted in a new casting in which the weight was 

 reduced by half, while at the same time the bursting strength 

 was doubled. The method used in arriving at this result is an 

 interesting illustration of sensible mathematical idealization. 

 The casting was regarded as an elliptical cylinder under hydro- 

 static pressure. As the stresses for this idealized structure were 

 already known, the design problem reduced at once to the 

 simple matter of establishing thicknesses sufficient to withstand 

 these stresses. 



Another example from the field of geophysical pros- 

 pecting is furnished by Mr. Eugene McDermott, Presi- 

 dent of Geophysical Service Inc.: 



(t) A specific case of mathematical research in instrument 

 design was recently encountered. The instrument in question 

 was intended for the measurement of gravity. After the machine 

 had been completely built it was found to be unexplainably 



inaccurate. After weeks of trial and error it was turned over 

 to a mathematician to try to find the trouble. He soon showed 

 by simple trigonometry that the axis of the instrument would 

 have to be located on its pivot with an accuracy which is not 

 attainable. He also pointed out a means of avoiding this 

 feature by a relatively simple change in design, and this appears 

 to have remedied the trouble. 



Another illustration from the petroleum industry, 

 but this time concerned with the production of oil 

 rather than prospecting for it, comes from Dr. E. C. 

 Williams, Vice President in charge of research of the 

 Shell Development Company: 



0) The petroleum industry has one important problem not 

 found in other fields; it has to do with oil production from the 

 ground. A mathematical problem arising from this subject is 

 the following: The oil-gas mixture underground flows under 

 pressure through porous media; with a certain spacing of wells, 

 determine the most economical way to recover this mixture. 

 This is sometimes equivalent to asking: "In what way can the 

 largest fraction of the oil be obtained over a certain period of 

 time?" Simplified problems of this kind have been solved by 

 potential theory methods, since classical hydrodynamics be- 

 comes too involved, and in the general problems where the 

 flow constants vary with liquid-gas composition, etc., partial 

 differential equations are found which can be solved by approxi- 

 mate methods. On the basis of the solution of this mathematical 

 problem, aided by extensive laboratory determinations of the 

 required constants, one is able to find the best of several ways of 

 producing from a given oil field. 



As a final example under the heading of economy, we 

 may mention the flight testing requirements imposed 

 upon the aircraft industry by the Civil Aeronautics 

 Authority. Of these, Mr. E. T. Allen, Director of 

 Flight and Research of the Boeing Aircraft Company, 

 says: 



(fc) It was formerly required that each type of transport 

 plane must be tested at all the altitudes at which it was intended 

 to be flown, and at all flying fields where it was expected to be 

 used. The cost of such testing was extremely high. A mathe- 

 matical study of steady flight performance has, however, 

 identified the basic parameters and established their relations to 

 one another. This has made possible a scientific interpretation 

 of flight test data taken at any suitable location convenient to 

 the aircraft factory, and a reliable conclusion therefrom as to the 

 performance to be expected under other conditions. This has 

 greatly reduced both the cost and the time necessary to establish 

 performance figures. 



Fifth: Sometimes experiments are virtually impos- 

 sible, and mathematics must fill the breach. An 

 example comes to me from Mr. Hall C. Hibbard, Vice 

 President and Chief Engineer of the Lockheed Aircraft 

 Corporation: 



(0 An unfortunate phenomenon that must be dealt with in 

 aircraft design is a type of violent vibration which may be set 

 up in the wings if the plane is flown too fast. It is known as 

 flutter, and is highly dangerous, since the vibrations may be of 

 such intense character as to cause loss of control or even struc- 

 tural failure. The technical problem is therefore to be sure that 

 the critical speed at which flutter would occur is higher than at 

 any at which the craft would ever be flown. It is a phenomenon 



