Simulation of the Saturn V Rocket on Electronic Analog Computers

English translation of the original German article: “Nachbildung der Saturn V-Rakete auf elektronischen Analogrechnern” by Dipl.-Ing. D.L. Teuber, Hermann Oberth Society (Hermann Oberth-Gesellschaft) Conference Proceedings No. 29, 13th Rockets and Spaceflight Conference, Darmstadt, 25–28 June 1964.

Summary

The development of large space vehicles of the Saturn V class encounters numerous problems that do not arise in the development of smaller rockets. The increase in effort required for an investigation becomes apparent when the influence of atmospheric disturbances on nonlinear control processes during the propulsion phase of the lowest stage of the spacecraft must be predicted in advance. Because of the uncertainty in predicting atmospheric disturbances, methods of probability calculus must be used. The high-speed analog computer GPS in repetitive operation is particularly suited to statistical investigation, since thousands of solutions for the overall behavior of the Saturn V can be evaluated.

The systems of differential equations describing the behavior of the Saturn V are presented. Taking into account fuel sloshing and bending oscillations, a system with 12 degrees of freedom is simulated on the analog computer. The stochastic input variable for the simulated equations is the wind velocity as a function of flight time. Magnetic tapes storing measured winds over several years are used for this purpose.

In another method, the stochastic input variable is generated by a noise generator and suitable time-varying shaping filters. The basis for this is the intensity and power spectrum of wind profiles. Exceedances of pre-set maximum values — such as bending moment, engine deflection, or angle of attack — are registered by the analog computer, enabling optimization by varying freely selectable control parameters.

Fundamental procedures for programming on the analog computer are described and typical control processes of the Saturn V are presented.

Introduction
The GPS Electronic Analog Computer
On Programming
Stochastic Input Variables
Typical Solutions for the Saturn V
Collaboration of Various Groups
Concluding Remarks
References
Figure Captions

1. Introduction

The size of spacecraft of the Saturn V class entails a considerable effort in the advance calculation of flight behavior. Problems arising, for example, from fuel sloshing and bending oscillations play a decisive role in the design of the control system. For the large number of calculations required, the analog computer proves to be an indispensable tool. On the one hand, the problems of flight stability are mathematically described by systems of ordinary linear and nonlinear differential equations, and the advantages of the analog computer can be exploited through the required integrations and differentiations. On the other hand, the limited accuracy of the analog computer is sufficient for an engineering treatment of the dynamic systems of large spacecraft.

The goal of simulating the Saturn V (Fig. 1) on the analog computer is to quickly gain insight into the temporal processes during flight and to estimate the influence of parameter changes. These investigations are supplemented on the digital computer with greater accuracy once the problem domain has been delimited.

Of particular importance, however, is the role of the analog computer in the investigation of stochastic input variables acting on the spacecraft. These include fluctuating aerodynamic loads on the launch pad, atmospheric turbulence and its influence on bending stresses, as well as turbulence and diffusion processes in boundary layers during flight. All these processes share methods of probability calculus and the evaluation of a large number of data.

In a publication by Rheinfurth (6), theoretical foundations were presented for a mathematical analysis of linear systems with stochastic input variables with regard to the behavior of spacecraft in atmospheric turbulence. The following text provides insight into the practical implementation of a statistical evaluation of atmospheric disturbances during the flight of the Saturn V.

2. The GPS Electronic Analog Computer

The GPS analog computer (manufactured by GPS Instrument Co. in Massachusetts; GPS stands for “General Purpose Simulator”) is designed for the task of delivering thousands of solutions for the overall behavior of the Saturn V in a short time. Through a time transformation, the processes on the analog computer run 3,000 times faster than in reality. This means that with an integration factor of 3,000 and a maximum amplifier voltage of 50 volts, the time span for one repetition cycle in repetitive computing must not exceed 150 milliseconds. For the flight of the Saturn V rocket from launch to engine cutoff of the S-IC stage, up to 50 solutions per second can be obtained in this way in repetitive operation. Due to the high computing speed, the computing amplifiers have a bandwidth of 1 MHz, and the phase shift in the operating range between 0 and 20 kHz is less than 1°.

One advantage of repetitive computing is that the solution curves are immediately visible in their entire course. Parameter variations in optimization tasks can thus immediately be recognized in their effects.

Fig. 2 shows the complete computing installation. Its most important computing elements include approximately 50 integrators, 50 summers, 350 potentiometers, 20 parabolic multipliers, and 15 function generators with 70 computing amplifiers. Time-varying quantities — such as aerodynamic values or stochastic functions — can be fed into the analog computer from 2 magnetic tape units, each with 7 tracks, at a tape speed of 1.5 m/s, synchronized with the computing processes. Conversely, solution functions can also be stored on the magnetic tapes. After analog-to-digital conversion, the solution functions can be used in a digital computer for further operations.

For statistical investigations, the installation also contains noise generators, switches, comparators, probability distribution analyzers, adjustable high-pass and low-pass filters, and counters. Figs. 3 and 4 show details of the installation. As output devices, in addition to the magnetic tapes, either 40-cm oscillographs are used, or the solutions can be photographed on a conventional oscillograph. A loop oscillograph of the Honeywell “Visicorder” type with a paper speed of up to 2 m/s can also be used.

The computing sequence as well as the input and output devices are centrally controlled by a generator. All synchronizing signals necessary for operation of the installation come from this generator.

3. On Programming

In its simplest form, the behavior of the Saturn V to be simulated is described by four linear differential equations with constant coefficients (2). It is assumed that attitude control in three mutually perpendicular axes is independent of each other, that no elastic deformations of the Saturn V or fluid oscillations in the propellant tanks occur, and that all nonlinearities can be neglected.

From the equilibrium of all forces:

Ẑ = k₁ψ + k₂α + k₃p

From the moment equilibrium:

ψ̈ + c₁α̇ + c₂p = 0

For the angular relationship:

α = ψ - Ż/V + w

For the control law:

p = a₀ψ + a₁ψ̇ + b₀α

Where:

Z = lateral deviation (relative to inertial reference frame)
ψ = angular deflection
α = angle of attack
αw = angle of wind attack
p = engine deflection
V = velocity vector of the rocket
C₁, C₂ = aerodynamic coefficients
k₁, k₂, k₃, a₀, a₁, b₀ = coefficients for force equilibrium and gains

Two of the roots of the characteristic equation of the third order can be associated with the rapid oscillation of the rocket about its center of mass. With good approximation, the oscillations are described by a second-order system with:

ωₙ² = C₁ + C₂(a₀ + b₀)

and: 2ζωₙ = a₁C₂

For the Saturn V, the control frequency is 0.2 Hz with a damping of 70% of critical. Both quantities are kept practically constant by varying a₀, b₀, a₁ during flight time.

Nine coefficients change during flight time: k₁, k₂, k₃, V, C₁, C₂, a₀, a₁, b₀.

Typical values for the time-varying coefficients are shown in Fig. 5. The input variable for the equations is the wind velocity V_w as a function of flight time. Negative values of C₁ indicate that the center of mass of the rocket lies behind the aerodynamic force application point, so that the uncontrolled rocket would be unstable.

Following a suggestion by Horn (4), it is practical to combine some of the variables as:

p = (C₁ + C₂a₀)/ω²

Advantages of this representation lie primarily in the clarity of results when the influence of parameter changes on transient processes is to be investigated. Fig. 6 shows the signal flow diagram including the frequency response F₀ of the actuator of the rocket engines.

The representation for the analog computer shows how the initially assumed constant parameters are assigned to various potentiometers, and so the bending moment M_t at a specific location of the Saturn V can be measured as a function of flight time and as a function of parameter changes.

A further advantage of combining aerodynamic and control engineering data into ωₙ and p is that multipliers on the analog computer can be saved. The time-varying parameters are shown in Fig. 7. Compared to Fig. 5, there are now 6 variables and thus 6 multipliers to be considered in the simulation on the analog computer. An approximation of the functions by diode function generators is possible with sufficient accuracy.

The basic circuit for a simulation on electronic analog computers is shown in Fig. 8. It is the extension of the circuit of Fig. 6 to time-varying coefficients. In this simple form, the analogy corresponds to a linear dynamic system with time-varying coefficients. The superposition principle also applies to systems of this type; however, unlike a system with constant coefficients, the transient response depends on the moment at which the input variable is applied. It is noted in (3) that the analysis and synthesis of such systems, described by linear differential equations with variable coefficients, are unsatisfactory from the control engineer’s standpoint and are described as a promising area for further research.

The circuit in Fig. 8 is merely the starting point for a general simulation of the Saturn V on the electronic analog computer. When fuel oscillations, bending oscillations, frequency responses of control system transfer elements (such as angle-of-attack sensors or accelerometers), or nonlinearities such as saturations, limits, or threshold sensitivities are taken into account, the representation of Fig. 8 quickly grows into a complex system. For the complete simulation of the Saturn V on an analog computer such as the GPS described above, several hundred computing elements are interconnected.

The equations of motion of the rocket can be represented as a system of linear differential equations with variable coefficients of the form:

[M(t)]ẍ + [C(t)]ẋ + [K(t)]x = [q(t)]

Where M(t), C(t), K(t) are the time-dependent mass, damping, and stiffness matrices of the system, {x} is the state vector, and q(t) are the generalized forces.

The equations of motion must be supplemented by the control law, which links engine deflection to the state vector. In the output equations for programming, the second derivatives of some variables appear in equations for the second derivatives of other variables. Difficulties can arise in the direct simulation of the output equations on the analog computer through closed algebraic loops. Such circuit loops contain only summers and tend toward instability at certain values of loop gain due to non-ideal amplifier characteristics.

This difficulty was initially circumvented by suitable eliminations in the output equations. A digital matrix transformation yields:

ẍ = -[M(t)]⁻¹[C(t)]ẋ - [M(t)]⁻¹[K(t)]x + [M(t)]⁻¹[q(t)]

This yields equations where each contains at most one second derivative. Fig. 9 shows the simulation of one of the 12 differential equations resulting from this elimination of second derivatives.

In total, approximately 300 potentiometers are needed for the complete simulation for the coefficients that can be assumed constant. All 20 multipliers simulate the most important time-varying coefficients.

A disadvantage of eliminating second derivatives using matrix transformation is that new coefficients are formed that are generally sums and products of those in the output equations. This causes the analog simulation to lose the clarity of the coefficients of the output equations. Each coefficient in the output equations has a specific practical significance for the rocket design, such as thrust, mass, or aerodynamic force. For this reason — which is significant when parameters are changed — a direct simulation of the Saturn V output equations was also carried out on the GPS analog computer by stepwise extension of the circuit of Fig. 7, including 4 elastic bending modes, 3 linear oscillator models for describing propellant oscillations in the booster tanks, and the frequency responses of various transfer elements. No stability difficulties arose in the computing circuit when the coefficients are not changed too far from the Saturn V’s nominal values. For wide parameter variations in general studies of large spacecraft, however, circuit stability can only be ensured through the described elimination of closed algebraic loops, at the cost of simulation transparency.

4. Stochastic Input Variables

Simulation of the dynamic processes occurring during flight of the Saturn V by electronic circuits with an analog computer is carried out at the G. C. Marshall Space Flight Center for various purposes and on various systems. The scope of simulation is determined by the size of the installation, i.e., the number of available computing units. In real-time simulation — as in the Astronics Laboratory — flight components such as amplifiers, filters, or hydraulic systems of the rocket are included in the simulation. For the investigation of adaptive controls, the Computation Laboratory uses mixed analog-digital computers.

All these simulations are based on the differential equations treated so far, which mathematically describe the dynamic processes, and on the programming methods for constructing the electrical circuits.

The special features of the GPS analog computer for simulating the Saturn V in the Aero-Astrodynamics Laboratory are now highlighted. During ascent through the atmosphere, the Saturn V is subjected to strong atmospheric disturbances. By their nature, these disturbances are stochastic. Mathematical statistics enables a description of the processes using amplitude squares and average exceedance rates, with correlation and power density functions (7, 1). System-theoretic methods fail, however, in the analysis of a nonlinear system such as is involved in a deeper study of the Saturn V. On the analog computer, however, treatment of these systems is possible without fundamental difficulties. It is now necessary to simulate not only the nonlinear differential equations but also the stochastic input variables.

Two methods are used for generating wind as a stochastic input variable. In the first method, magnetic tapes are used on which actually measured wind data up to an altitude of 21 km are stored. The wind data were obtained at Cape Kennedy by radar measurements of balloon ascents. The data are statistically averaged over a 3-second period. At a balloon ascent speed of 8 m/s, the measurements yield wind velocities at height intervals of 25 meters. The high computing speed of the GPS analog computer allows daily measurement data from several years to be played back and statistically evaluated as stochastic input variables for the Saturn V simulation in a few minutes. In this way, the behavior of the Saturn V under atmospheric disturbances can be measured. Maximum values can be assigned to the variables of interest — such as engine deflections or angle of attack during Saturn V flight — on the analog computer. The number of exceedances of these maximum values is automatically registered by the computer and can be related to the total number of wind profiles.

Of particular importance is the repeated playback of a specific wind profile on a closed magnetic tape. Through variation of freely selectable control parameters, certain optimization procedures can be applied with the aim of achieving a good compromise between often mutually contradictory design criteria.

Optimizing transient processes for a specific wind profile does not necessarily mean that the behavior of the Saturn V will be satisfactory for all winds probable at the launch time. Therefore, another method is currently being used for generating wind as a stochastic input variable for the dynamic equations of the Saturn V. The wind is generated using the circuit schematically shown in Fig. 10. A noise generator delivers a stochastic, normally distributed time function with a practically constant power density. This white noise is described by a Gaussian distribution. The frequency range over which the power density is constant is larger than the bandwidth of the analog computing circuit. The wind is then divided into three components: a monthly mean value, daily fluctuations, and turbulence. This division is arbitrary but practical for measurement engineering and for a synthesis in which each component can be varied independently of the others. The monthly mean value is generated by a function generator; daily fluctuations and turbulence are filtered from the white noise using time-varying filters. The bandwidth of the filters F_t and F_d is expanded according to the flight speed of the rocket. The average number of zero crossings, distribution, and exceedance of specified values per unit time in daily fluctuations and turbulence are assigned to the filter parameters θ and ε (5).

The main purpose of this statistical analysis is threefold. First, a statistical description of measured winds is to be carried out with regard to investigating the dynamic processes during the flight of the Saturn V. Electronic circuit simulation of the wind is an essential tool for this. Second, complex systems such as the differential equations of the Saturn V cannot be investigated with analytical methods. Noise generators and analog computers are indispensable tools for a realistic investigation of these systems. Third, the influence of parameter changes on the transient processes of the Saturn V with stochastic input variables is to be investigated.

For all these statistical investigations, a large number of solutions is needed, which the GPS computer can deliver in a short time. An evaluation of these solutions is carried out by an electronic correlator, using a magnetic tape device as a delay device. Using correlation functions, relationships are described — for example, between measured and “synthetic” winds, between loads and deflections during flight with atmospheric disturbances. These investigations are currently in full progress.

5. Typical Solutions for the Saturn V

Fig. 11 shows typical wind velocity profiles, excluding turbulence, as they can occur during the flight of the Saturn V. For simplified analog simulations, the time-varying coefficients are “frozen.” The investigation is then valid for a flight duration of a few seconds — for example, near maximum dynamic pressure at the 80th second of flight. A mean flight velocity of approximately 500 m/s is assumed in an altitude range between 12 and 15 km. For investigation of events around the 80th second of flight, the wind profile designated C (constant coefficients) is used.

For time-varying coefficients, the wind velocity as a function of altitude is converted to wind velocity as a function of flight time, taking into account the varying flight speed. The curve designated TVP (time varying parameters) equals the curve labeled C at the 80th second.

Fig. 12 shows typical transient processes for the Saturn V. On the one hand, the coefficients frozen at the 80th second were used in the dynamic equations for the wind velocities labeled C in Fig. 11. On the other hand, the time-varying coefficients shown in Fig. 7 were programmed and the wind velocities labeled TVP in Fig. 11 were used. The agreement of the corresponding variables at the 80th second is quite good. At this point, the variation of coefficients is slow compared to the transient response and to the time constants of the entire system. At other points in time, the coefficients change so rapidly that they can no longer be assumed frozen for recording the transient behavior during a specific time period.

As an example of a statistical evaluation, Fig. 13 shows the average number of exceedances for various values of bending moment at a specific location of the Saturn V when time-varying coefficients are applied. Analytical methods fail when nonlinearities in the control loop are included. The analog investigation is particularly valuable, as the distribution functions of interest for system behavior can easily be obtained by measurement.

For the distribution function in Fig. 13, the measured winds of the months January, February, and March of 1958 from Cape Kennedy were used. With 2 measurements daily, 180 wind profiles are available. Because of the extremely short computing time of individual calculation runs on the GPS, the time required to measure such a curve is small and a matter of a few minutes. The distribution functions for several variables can be determined simultaneously. On the GPS analog computer, measurements are automatically made with comparators and counting circuits during playback of the magnetic tapes.

6. Collaboration of Various Groups

The analog computer brings together the design data of many departments. The results of the analog simulation in turn influence the practical design of the Saturn V. This interaction is described more closely through some examples.

Good collaboration with the group conducting wind measurements is essential for successful execution of the analog measurements. The accuracy and resolution of wind measurements must be known for error investigations in analog studies, which in turn requires precise knowledge of wind measurement methods. In addition to radar tracking of balloon ascents and optical measurement of the time behavior of smoke columns, further wind measurement methods are currently being investigated for their usefulness. These include radar backscatter measurements, laser methods, and others.

Of course, the method of generating wind using electronic circuits for analog measurements builds on wind measurements. Both methods are linked by a mathematical description using statistics.

A number of time-varying coefficients originate from the aerodynamics of the spacecraft. Since aerodynamic forces generally depend on Mach number, angle of attack, and configuration of the flight body in a very complicated way, these are determined by wind tunnel investigations on models. The analog investigations show their influence on the dynamic behavior of the Saturn V including a control system.

In a dynamic test stand, the Saturn V is suspended to obtain the elastic behavior of the entire structure. The rocket is moved laterally and longitudinally at the engine attachment point with time-sinusoidally varying forces. The investigations are conducted in three mutually perpendicular planes and with various simulated propellant loads. From measurements and calculations, a model of the bending oscillations is developed. The natural frequencies and damping values for the mathematical description of these extremely lightly damped oscillation systems are reflected on the analog computer.

In a similar way, propellant oscillations are determined on a test stand. On the analog computer, the data determined from measurements are used to make predictions for the behavior of these propellant oscillations in combination with other dynamic processes.

Furthermore, filters can easily be simulated on the analog computer that must have a specific frequency response for stabilizing the closed control loops of the Saturn V. These analog computer investigations feed back into the practical design of the overall control system.

Particularly close collaboration exists between flight evaluation and analog measurements. Actual flight data for the Saturn V will only be available in the future. Current analog investigations, however, build on the existing flight data of the Saturn I. The good agreement between actual flight behavior and the behavior predicted by the analog computer for past flights justifies great confidence in the realization of current predictions for the Saturn V based on the analog model.

7. Concluding Remarks

The analog computer, as a mathematical instrument, is a link between theory and practice and an indispensable tool for simulating the Saturn V. This simulation extends theory — for example, in the attempt at a mathematical description of processes occurring when stochastic variables act on systems of ordinary nonlinear differential equations with variable coefficients. Because of its high computing speed, the GPS analog computer in repetitive operation is suited to deliver, in a short time, the evaluation of a very large number of solutions necessary for deeper penetration into theory.

For practical purposes, the analog computer quickly provides a vivid insight into the complex processes that can occur during the interaction of propellant oscillations, bending oscillations, and control processes during the flight of the Saturn V. Through parameter changes and optimization procedures, the design criteria can be found that guarantee a successful launch of the Saturn V.

References

Bieber, N. E., “Missile Structural Loads by Nonstationary Statistical Methods,” Journal of the Aerospace Sciences, April 1961.
Geissler, E. D., “Problems in Attitude Stabilization of Large Guided Missiles,” Aerospace Engineering, Vol. 19, No. 10, October 1960.
Gibson, J. E., Nonlinear Automatic Control, McGraw-Hill Book Co. (year not stated in text).
Horn, Teuber, “A General Analog Simulation of Large Space Vehicles,” (in preparation, NASA Technical Note).
Rainal, A. J., “Zero-Crossing Intervals of Gaussian Processes,” IRE Transactions on Information Theory, October 1962.
Rheinfurth, M., “Das Verhalten von Raumfahrzeugen in atmosphärischer Turbulenz” [The Behavior of Spacecraft in Atmospheric Turbulence], 12th Rockets and Spaceflight Conference, 1963, Hamburg, Germany.
Wiener, Norbert, Extrapolation, Interpolation and Smoothing of Stationary Time Series, New York, 1950.

Figure Captions

Fig. 1: The Saturn V LOR Rocket
Fig. 2: The GPS Analog Computer, complete installation
Fig. 3: The GPS Analog Computer, magnetic tapes and statistical computing units
Fig. 4: The GPS Analog Computer, control panel and computing units
Fig. 5: Time-varying parameters
Fig. 6: Signal flow diagram and analog computer diagram for constant parameters
Fig. 7: Consolidation of some variable parameters (see Fig. 5 for dimensions)
Fig. 8: Analog computer diagram for time-varying parameters
Fig. 9: Typical simulation of a second-order system
Fig. 10: Circuit for wind simulation
Fig. 11: Typical wind velocities for constant and time-varying parameters
Fig. 12: Typical transient processes of the Saturn V for constant and time-varying parameters
Fig. 13: A distribution curve

[Translation covers the first 24 pages (main text and figure captions); the original continues with figures/charts for approximately 19 more pages — full translation of remaining pages deferred as they consist primarily of diagrams and tables.]