Appendix - Part E

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Appendix

B. Analysis

    The so-called Technical System of Units using meter for length, second for time, and kilogram for weight, is the one most widely used today. The International System of Units (Système International d'Unités), which uses the physically correct unit of kilogram mass instead of the kilogram weight, although generally accepted, is not yet in truly general use.  [p.159]

    In order to avoid complicating the reading of this book by the use of terms familiar to only a small portion of the readers, I have used the Technical System throughout. The corresponding figures of the English system (ft. lb, etc.) are added where meaningful.

2. Procedure

    The analytical investigation centers around the formula

which defines the conditions for a one-stage flight from earth into a desired orbit and in which

    W₀ = lift-off weight, in kg (or lbs) (total weight at the beginning of the flight)

    W_c = weight at the end of the flight, in kg (or lbs)

    e = the natural number = 2.71828

    v =    velocity at the end of the flight, in m/sec (or ft/sec)

    g =    acceleration of gravity = 9.81 m/sec² (or 32.2 ft/sec²)

    I_sp=    specific impulse, in sec

    In the case of a new vehicle development the first two values are the unknowns. All other figures are either known or can be estimated. In order to eliminate one of the two unknowns, they are usually coupled by introducing the value

    In this expression, W₀ - W_c represents the weight of the propellant consumed during the flight, and m indicates, therefore, the ratio of propellant weight to total weight. Since the proper definition is not in weight but in mass, m is called the mass fraction. For rockets of conventional construction or such with minor deviations, the range of the expected mass fraction is either known from experience or can be obtained by relatively simple extrapolation. The introduction of the mass fraction is therefore a very advantageous means to solve the basic equation. In the case of a subsequent development and hardware program these figures are watched closely, of course, and corrections are made when necessary.

    In the present case, however, there was no way to estimate the mass fraction with an even remotely satisfactory degree of accuracy. On the other hand, it is clearly possible to obtain weight estimates of the subsystems by introducing their characteristic properties as parameters. Since it was intended from the outset to include in the investigation a wide range of the specific impulse, that value too was treated parametrically, as was, finally, also the lift-off weight over the expected range.

    The easiest way to handle these many variables is a subdivision of the analysis into two parts, each of which using the lift-off weight W₀ as reference value. The first part determines the weight relations defined by the flight conditions as expressed in the formula above. In the second part, structural weights are determined as fractions of W₀. Since both parts of the analysis include the full range of the specific impulse, it is possible to superimpose their results directly. This procedure is done graphically. The points of intersection of respective curves so obtained evidently satisfy the conditions of both parts of the analysis, and are therefore part of the final result. The connection of these points to a new curve renders the graphic presentation of the end result. The accuracy of the. procedure is sufficient, because it is higher than that of the underlying assumptions. These assumptions will now be discussed.  [p.160]

    (a) Flight conditions