Methods of Design Analysis

The computer program would have defined the minimum cost vehicle and its reliability as well as the weight, cost and reliability of each subsystem, and the system life cycle cost. This methodology made the critical assumption that the required relationships are continuous and could be prepared even for new forms of construction, materials, etc. Moreover, the procedure would have given designers little visibility. Computer solutions, checked by design layout, would require repeating the entire procedure. Defining a practical methodology was viewed as a perplexing, unresolved problem.

Faced with the task of designing a space launch vehicle to the minimum cost design criteria in a period of several months, a simplified method of design analysis was quickly devised. (A deadline is a superb stimulator!) It permitted stages and payloads to be treated as separate entities, thus breaking down the design problem into more manageable proportions. The basic procedure is explained in Figure 1 in a simplified manner by ignoring the many details involved.

Figure 1. STAGE, LAUNCH VEHICLE, PAYLOAD OPTIMIZATION PROCEDURE

Consider Curve A to represent a family of cost-optimized first stages that achieve a given burnout velocity and reliability in terms of cost and the "payload" weight the stage carries, where "payload," in this instance, is defined as all weight above the first stage. The optimum size of the first stage is determined in conjunction with the optimum sophistication of the second stage. The second stage is depicted as Curve B. Curve B represents the cost of the second stage plotted against , where  is the ratio of stage burnout weight to used propellant weight.

The optimum size of the first stage and the optimum degree of sophistication of the second stage occur when the local slopes of Curves A and B at the same payload weight are equal and opposite or, in effect, when the sum of Curves A and B is minimum. The positive value of the slope is defined as K, and it is the value of a pound of weight in the second stage. As explained later, K is used to permit finalizing the second stage design in an iterative manner. The same procedure applies to all higher stages and payloads. Note that the value of K increases with each, successive stage.

Thereby Curve A can represent two or more stages for the purpose of designing the next, upper stage or the final stage and payloads. This infers that payloads are designed to the same sophistication level as the final stage. It is also noted that the value of a pound of payload weight for design purposes is always somewhat less than launch vehicle cost in $/lb of payload.

Parametric analysis, employing current values for weight, cost and reliability are generally used to identify the "ballpark" design. The design is then defined through iteration using the breakeven relationships between the various design paramters. Breakeven relationships are relatively simple to derive for a first stage, and are obtained by taking partial derivatives; however, the same procedure proved unmanageable in treating higher stages.

In treating higher stages only the breakeven relationships are between cost and weight, and specific impulse and propellant cost are considered, since the values for cost and weight should reflect tradeoffs with reliability previously conducted. Consider the cost-weight breakeven relationship that is universally applicable and is related to K. By inspection, when w_B > w_I:

where,

w_B = breakeven weight. lbs
w_I = initial weight, lbs
c_B = breakeven cost, $/lb
c_I = initial cost, $/lb

In Figure 2 the above equation is plotted for a large range of c_I / K values. The figure shows that when the ordinate parameter is about one, there is a good likelihood that it might pay to decrease weight of the ballpark design in order to decrease cost. Furthermore the figure shows that when the parameter is near 100, which generally corresponds to relatively costly and low-weight components, it would probably pay to increase weight almost indiscriminately to effect a cost reduction.

Thus a designer of a subsystem need only be given the value of K and the reliability goal in order to finalize a minimum cost design. In achieving the reliability goal he also may have to simultaneously refine the cost-reliability and cost-weight tradeoffs, perhaps with the help of specialists in the area of reliability.

FIGURE 2. HARDWARE COST-WEIGHT BREAKEVEN RELATIONSHIPS

Such analytical simplicity was not achieved with the specific impulse versus propellant cost breakeven relationship. The introduction of the K parameter, however, did reduce the complexity of the partial derivatives. The relationship, applicable to any stage, may be found in The Aerospace Corporation report, "Proposed Minimum Cost Space Launch Vehicle System," by A. Schnitt and Col F.W. Kniss, July 1968, p 2-17. 

At times it may be more convenient to determine the optimum hardware in a manner described schematically in Figure 3, particularly in the design of payloads. Figure 3 displays an array of designs of a subsystem or component, each having the same performance and reliability.

FIGURE 3. TECHNIQUE FOR SELECTING COST-OPTIMUM HARDWARE DESIGN

Design A may represent the minimum weight, current aerospace industry type of hardware while Design E may be called the "cost-end-point" design since its main consideration is minimum cost though it may be heavy and large in volume. Generally, several intermediate designs are feasible and can be sketchily defined.

It is suggested that the designer display on the graph the negative value of K previously determined for the stage or payload under design iteration. (See Figure 2.) By so doing, the designer might be able to more closely estimate the location within the spectrum (Designs B, C and D) where further design efforts should be concentrated. This technique is a graphical presentation of the breakeven equation previously defined. The cost saving in comparison with the minimum weight design may be estimated as illustrated.

The previous approach to payload design infers that there is a close match with an available launch vehicle of the required weight capability. If this is not the case, and the singular launch vehicle has a larger or smaller payload capability by a significant amount, the optimization parameter K can be raised or lowered to fit the launch vehicle capability. By employing this technique it is evident that a more cost-balanced payload design would cost less.

Consider another payload/launch vehicle mismatch condition. In this case the singular launch vehicle can carry multiple payloads. The payload optimization procedure could employ the full value of K as determined by the launch cost per pound of payload without incurring a significant loss in being off-optimum.

In the case when there is a singular launch vehicle available to carry a singular payload, there is no apparent payload design cost-optimization procedure. It is suggested, however, that the major, more important components of the payload be designed first, using the full value of K as determined by the launch cost per pound of payload. Any payload weight still available would then accommodate the other payload requirements. Never, or hardly ever, should the cost of the main components be raised in cost and reduced in weight to make room for the lesser payload objectives. An easily prepared analysis would be able to prove this.
 

How do today's computer capabilities affect space launch vehicle design analysis? [clicking on the above opens link to discussions that followed the initial posting of this column]

Next Column: Design examples

Return to Published Columns