Note 25 — Chapter VI · BALANCING THE AERODROME

The following formulæ proposed by Mr. Chas. M. Manly show how the center of pressure may be moved any desired distance either forward or backward without in any way affecting the center of gravity, and by merely moving the front and rear wings the same amounts but in opposite directions, the total movement of each wing being in either case five times the amount that is desired to move the mean CP1, and the direction of movement of the front wing determining the direction of movement of CP1.

In Figure 7, CP_fw and CP_rw are the centers of pressure of the front and rear wings respectively; the weights of the wings, which are assumed to be equal and concentrated at their centers of figure, are represented by w, w, and a is the distance of the center of pressure in either wing from its center of figure. The original mean center of pressure of the aerodrome is CP₁, W is the weight of the aerodrome, supposed to be concentrated at CG₁, while m is the distance from CP_rw to CG₁.

Now, if we have assumed that the rear wing, being of the same size as the front one, has a lifting effect of only 0.66, and on this assumption is calculated the proper relative positions of the front and rear wings to cause the CP₁ to come directly over the CG₁, and upon testing the aerodrome find that it is too heavy in front and, therefore, wish to move the center of pressure forward an amount, say b, without affecting the center of gravity, we can calculate the proper relative positions of the front and rear wings in the following manner. While the aerodrome as a whole is balanced at the point CG₁, the weight of the wings is not balanced around this point, for the rear wing, owing to its decreased lifting effect, is proportionately farther from CP₁ than the front wing. In order, therefore, to avoid moving the center of gravity of the machine as a whole, any movement of the wings must be made in such a way as to cause the difference between the weight of the rear wing multiplied by its distance from CG₁ and the weight of the front multiplied by its distance from CG₁ to equal a constant: that is,

w(m + a)−w(0.66m−a) = constant,

and

0.33wm + 2wa = constant.

FIG. 7.

FIG. 8.

FIG. 9.

FIGS. 7–9. Diagrams Illustrating formulæ for moving C. P. without disturbing C. G.

If now the wings be moved so that CP₁ is moved forward a distance b, we may indicate the distance from CG₁ to the new CP_rw by z, and equating the difference between the weight of the rear wing multiplied by its new distance from CG₁ and the weight of the front wing multiplied by its new distance from CG₁ and making this difference equal to the constant difference, we can calculate z in terms of m and b, as follows:

Fig. 8,

w(a + z)−w(0.66(z + b) + b−a) = 0.33wm + 2wa,

∴ z = m + 5b.

Knowing z, we readily find that the new distance from CP_fw to CG₁ equals:

0.66(z + b) + b = 0.66m + 5b.

In a similar manner we may calculate the proper relative positions of the front and rear wings when we wish to move the center of pressure backward a distance, b, from the original CP₁ without changing the position of CG₁. From Fig. 7, we have as before:

w(m + a)−w(0.66m−a) = constant,

0.33wm + 2wa = constant.

Fig. 9,

w(z₁ + a)−w(0.66(z₁−b)−b−a) = 0.33wm + 2wa.

∴ z₁ = m−5b.

Similarly we have for the new distance from CP_fw to CG₁:

0.66(z₁−b)−b = 0.66m−5b.