in the preceding section on diffusion and leakage the appendix opening should be small. It can be seen that as we make one of our conditions better, we must sacrifice at least one of the others. Therefore, balloons must be designed compromising rate of rise, balloon thickness, and appendix opening. Methods of decreasing the appendix opening, except during the valving of lifting gas, are discussed in other sections of this technical report. In general they consist of means of applying a delicate relief valve, capable of opening to a large area with application of only slight internal pressure, and also closing tight upon release of this internal pressure.
G. A General Equation of Motion
If we collect and relate the variables incidental to balloon flight, we may form a general equation of motion. This is most easily expressed in terms of forces acting upon the balloon system. We may equate an acceleration term plus a drag or friction term against a term to include all other forces:
$$
(1) \quad m \frac{D^2 z}{g} + n (Dz)^2 = \sum F
$$
This is a differential equation of a type common in mechanical vibration problems, and solution for the variable $z$ would not be difficult if relationships of the many variables included in the terms $n$ and $\sum F$ were simple. However, the complexity of the balloon system introduces many terms as parts of $n$ and $\sum F$ .
We shall first state the more complex form of equation (1) above and then attempt to explain the variables included in each part of the equation. As will be shown, it is extremely difficult to find a complete solution of the equation since many of the variables are in themselves extremely complex and at this time incapable of accurate solution. Therefore, our discussion will be more of a qualitative rather than a quantitative nature.
The general force equation is:
$$
(2) \quad \frac{W}{g} \frac{D^2 z}{g} + C \frac{\rho}{2} A (Dz)^2 = V_b (\rho_a - \rho_g) - W \pm F_{atm}
$$
The force due to acceleration $F_A = \frac{W}{g} \frac{D^2 z}{g}$ where:
- $W$ = weight of the balloon system
- $g$ = acceleration of gravity
- $\frac{D^2 z}{g}$ = acceleration of the balloon system (An acceleration in the direction of greater altitude is considered positive.)
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