Page:Aerial Flight - Volume 2 - Aerodonetics - Frederick Lanchester - 1908.djvu/68

Chapter III

 

The Phugoid Theory.—the Flight Path Plotted

 

§24. Preliminary Considerations.—The form of equations (11) and (13) is such as to indicate that so long as a value of C is chosen proportional to the square root of H_n (the natural height), the form of curves for all values of H_n will be geometrically similar, though of different linear scale. This follows from the form of the equation and from dimensional theory, for the constant C is of the dimensions of the square root of a linear quantity. It is thus of no consequence what particular value of H_n be chosen for the purpose of plotting the curves; a series of curves plotted for any one value of H_n applies equally to any other value if read to an appropriate scale.

It has already been pointed out that the phugoid equation does not lend itself to plotting in the ordinary way; the form of the expression $\cos \Theta ={\frac {H}{8H_{n}}}+{\frac {\mbox{C}}{\sqrt {H}}}$ is one that, so far as the author is aware, is not susceptible of being reduced to co-ordinate form. It is consequently necessary to employ some other method of plotting, thus taking the equation for the radius of curvature (13),

$r={\frac {2}{{\frac {\mbox{C}}{H{\sqrt {H}}}}-{\frac {2}{3H_{n}}}}}$

and laying the curve off step by step by means of a trammel the difficulty is overcome.

 

§ 25. Plotting the Curves. The Trammel.—Having selected any convenient value^[1] for H_n or employing that proper to some

1. H_n = 61 in the plottings given. There is no real reason why this particular value of H_n should have been chosen; it could equally have been made = 1.

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