220 SYMPOSIUM ON AERONAUTICS. 



For some purposes it might be the variables w or w — for example, 

 the whole theory of gusts here given depends on the gust being 

 small and producing small effects, and if by an applied force, the 

 values of u or w should become too large, the theory would become 

 worthless. Again, if the question had to do with the strain on the 

 machine, the derivatives du/dt and dzv/dt would be the essential 

 objects of interest, and should be maximized. Finally it might be 

 the values x^fudt and z=^fwdt — the actual displacements of the 

 machine — which we desired to examine. Let us therefore consider 

 several problems seriatim. 



i6. Case i. — To maximize n with a head-on gust m^ = ^*p'. 



aip + n . , .I28z>+ .598 



u = — --, — e* 



A' .598 - p^^- ^.ogSip 



The maximum value of 



.1282^2^.5982 .i282(/)2 + 21.83) 



(.598 - p-y- + 4.0982;^2 pi ^ 15.59^2 _^ _3--5 



occurs when />- is o, that is, "resonance" occurs for p = o, the 

 amplitude of the force and the oscillation being the same. 



Case 2. — To maximize zv with a head gust. 



This was treated above (§ 10). The ratio .136 was found; the 

 required value of p was ."jy^i- 



Case 5. — To maximize u with an up-gust ii\. 



.i62pi 



11 = = p'^pi 



.598 - ^2 _|_ 4.098/pi • 



The condition is /> = .776 as is Case 2 ; the ratio is .04. 

 Case 4. — To maximize zv with an up-gust. 



3 -95^^ + -598 . . 



iu = — — e p 



.598 - p'^-{- 4.09Spi 

 The maximum value of 



3-95-p- + -598- 3-95'^/^- + -0228) 



(.598 - pr~ + 4-098-^- P' + I5-59P' + -3576 

 occurs when p^ = .022 and p = .i^, and the amplitude ratio is 

 about I. 



