Dynamics and Simulation of Flexible Rockets

By Timothy M. Barrows and Jeb S. Orr

Dynamics and Simulation of Flexible Rockets provides a full state, multiaxis treatment of launch vehicle flight mechanics and provides the state equations in a format that can be readily coded into a simulation environment. Various forms of the mass matrix for the vehicle dynamics are presented. The book also discusses important forms of coupling, such as between the nozzle motions and the flexible body.

This book is designed to help practicing aerospace engineers create simulations that can accurately verify that a space launch vehicle will successfully perform its mission. Much of the open literature on rocket dynamics is based on analysis techniques developed during the Apollo program of the 1960s. Since that time, large-scale computational analysis techniques and improved methods for generating Finite Element Models (FEMs) have been developed. The art of the problem is to combine the FEM with dynamic models of separate elements such as sloshing fuel and moveable engine nozzles. The pitfalls that may occur when making this marriage are examined in detail.

• Covers everything the dynamics and control engineer needs to analyze or improve the design of flexible launch vehicles
• Provides derivations using Lagrange’s equation and Newton/Euler approaches, allowing the reader to assess the importance of nonlinear terms
• Details the development of linear models and introduces frequency-domain stability analysis techniques
• Presents practical methods for transitioning between finite element models, incorporating actuator dynamics, and developing a preliminary flight control design

• Language: English
• Publisher: Academic Press
• Release date: Dec 10, 2020
• ISBN: 9780128199954

Timothy M. Barrows

Dr. Barrows has over 30 years of experience in analysis and simulation of complex mechanical systems for NASA and various agencies of the Department of Defense. His engineering expertise includes aerodynamics, multi-body dynamics, and simulation. A particular expertise is simulation of multibody systems. He has either directly created or supervised the construction of high fidelity simulations of several systems, including the attitude control of a satellite, a generalized robotic manipulator model, the space station mobile transporter, and the flight mechanics of precision guided airdrop systems. Other work has included successful airdrop tests of a gliding autogyro with folding rotor blades, and a concept definition of a large vehicle designed to fly in ground effect. He has served as Section Chief of the Dynamical Systems Group at Draper, in which capacity he served as the engineering task leader of for the Space Station Dynamic Interaction program. During the past ten years, Dr. Barrows has focused on the development of rocket simulations. This has included a wide variety of rockets for both private and government sponsors, culminating in work on NASA’s space launch system.

Book preview

book.

Chapter 1: Introduction

Abstract

The state of current literature on rocket dynamics is discussed. It is observed that there is a need for a consistent formulation that takes advantage of both finite element methods and the ability of computers to easily handle large matrices. There is a need for the dynamics and control engineer to create a model that meets the requirements of his or her particular rocket. Modeling choices during this activity are discussed. The notation system that is used throughout the book is explained.

Keywords

large rockets; Saturn V; integrated body model; reduced body model; notation system; column matrix; cross product matrix

Rockets, like most things, become more complicated as they grow larger. Judging from the similarity of external appearance, it might seem that going from a small rocket to a large rocket would be a simple extrapolation according to size. However, this is not the case. Some idea of the reason for the added difficulty can be obtained from the following quote from J. B. S. Haldane:

… consider a giant man sixty feet high – about the height of Giant Pope and Giant Pagan in the illustrated Pilgrim's Progress of my childhood. These monsters were not only ten times as high as Christian, but ten times as wide and ten times as thick, so that their total weight was a thousand times his, or about eighty to ninety tons. Unfortunately the cross sections of their bones were only a hundred times those of Christian, so that every square inch of giant bone had to support ten times the weight borne by a square inch of human bone. As the human thigh-bone breaks under about ten times the human weight, Pope and Pagan would have broken their thighs every time they took a step. This was doubtless why they were sitting down in the picture I remember. But it lessens one's respect for Christian and the Giant Killer.

In this example, increasing the bone cross section by a factor of a hundred is not enough – it must be increased by more than a hundred. In other words, the structural weight fraction must be increased. In the design of rockets, however, the mere suggestion of increasing the structural weight fraction will produce the most pained anguish. A good portion of this extra weight will be taken out of the payload. As a typical payload weight is less than ten percent of the total rocket weight at launch, it is easy to see how the payload can disappear entirely without a stringent effort to minimize the structural weight. The result is that the design of large rockets becomes an almost desperate effort to improve structural efficiency.

From a dynamic standpoint, as the scale increases, the rocket grows flimsier and flimsier. The natural frequencies of more and more flexible modes creep downward into a range that is within the control bandwidth. The opportunities for dynamic interaction proliferate. The control engineer must verify that all of these interactions are benign and stable. Doing this requires methods for constructing simulations that can efficiently deal with a large number of dynamic modes.

Perhaps the most famous large rocket ever built was the Saturn V of the Apollo space program. Since the time of that program, major advances have taken place in our ability to analyze structures using finite element methods. At the same time, modern computer tools such as MATLAB® have promoted the use of matrix techniques and made it increasingly easy to deal with large matrices. The purpose of this book is to provide a uniform foundation for modeling all these interactions that takes advantage of these developments.

The dynamics of an ascending rocket are typically presented for planar motion. That is, the resulting equations are valid for either the pitch plane or the yaw plane. This approach does not provide any insight into the possible coupling that may exist between motion in one plane and that in another. Such coupling may arise from asymmetries in either the mass distribution or the stiffness distribution.

The planar dynamics of a rocket can be found in many sources. These sources fall into two separate camps, which might called the reduced body approach and the integrated approach. The characteristic feature of the former is that the translation and rotation equations are written for a reduced body consisting of the rocket without the sloshing fuel mass. One example of this approach is the textbook by Greensite [1]. A comprehensive treatment of the planar motion of a rocket was developed for the 1960's Atlas rocket program, although the technical reports (and similarly, company reports that are cited elsewhere) are not available in the open literature. Related formulations were independently derived by Rheinfurth and Hosenthein [2]; these are presented in part in the compilation by Garner [3] and eventually appear, without reference, in the classic paper by Frosch and Vallely [4]. An early example of a derivation in the open literature is the work of Bauer [5]. He provides an analysis of a flexible rocket with sloshing fuel mass. His analysis does not include a gimbaled engine.

Rocket dynamics is essentially multibody dynamics applied to a system consisting of a rocket body, engine nozzles, and slosh masses. The multibody model must be coordinated with the structural dynamic model – they must both take either the reduced body or the integrated body approach. Thus if a finite element model already exists for the rocket, the dynamicist will have to go along with whatever approach was taken during the creation of that model. An integrated body finite element model, as the name implies, contains all of the mass of the rocket, including the slosh masses and engines. For the creation of this structural model, the slosh masses and engines are locked to the vehicle. Thus the relative motion of the slosh masses is not included, and the engine gimbal actuators are treated as rigid. The result of the finite element analysis is a set of eigenvalues (mode frequencies) and eigenvectors (mode shapes), which become input parameters to the dynamic model (the subject of the present treatise). In a reduced body finite element model, either the slosh masses or the engine masses, or both, are removed from the rocket, and a finite element model is created from what is left. Within the dynamic model, the effects of the relative motion of the slosh and engine masses are treated in different ways for the integrated body model and the reduced body model.

This book begins with the integrated body approach, which is derived in Chapter 2. As will be seen, the reduced body approach has the disadvantage that the results contain more terms. It turns out, however, that no guarantee can be provided that the mass matrix using the integrated approach is positive-definite. Indeed, it can be shown that if a sufficiently large number of modes are included, the mass matrix will become non-positive-definite. Thus the reduced body approach, while less convenient, is the safer of the two approaches. This is discussed in Section 2.6.

Besides the issue of the integrated body approach versus the reduced body approach, there are two other major decisions that must be made before embarking on the analysis of rocket dynamics. For preliminary studies, it is often assumed that the Thrust Vector Control (TVC) actuators are very stiff, such that the engine motion can be computed independently from the rest of the dynamics. In other words, engine motion is prescribed. Chapter 2 goes into this in some detail. For purposes of the present discussion, it is sufficient to state that one must either (a) assume a given engine motion, which acts like a disturbance to the motion equations, or (b) assume a certain actuator torque on the engine, in which case the state vector is expanded to include variables that specify the engine motion. A third decision must be made as to whether to model the slosh motion as a point mass that slides in a y-z plane at the end of a spring (the spring model), or to model it as a point mass on the end of a pendulum. Thus there are a total of eight possible outcomes from making these three binary decisions about the model formulation. For this reason, this book does not provide a final result for the system equations of motion, but rather attempts to present the results in such a way that the analyst can select the equations and terms for the particular formulation that is most appropriate.

Notation system

The analyses herein follow the system used by Hughes [6]. His system makes a distinction between a vector and a column matrix. A vector is a mathematical quantity with both magnitude and direction in three-dimensional space, and is independent of the system of coordinates used to express it.

Suppose there is a reference frame a and a reference frame b . A vector may be written using its frame a components

(1.1)

or using its frame b components

(1.2)

(Symbols in italics are scalars). Both expressions represent exactly the same vector. In frames a and b, the associated column matrices are expressed as

(1.3)

One feature of the Hughes system is that the superscript representing the frame is dropped. Thus it may be necessary to read the text to determent the frame in which each vector is expressed. This may make it more difficult to jump into the middle of a derivation and understand what everything means. However, this drawback is more than compensated by the fact that the notation is less cluttered. Appendix A contains a glossary of symbols that may be helpful in finding where each symbol is first defined.

is the coordinate frame of body 1. The statement "v is really a shorthand for the statement that v ."

indicates the time derivative of v indicates the time derivative of v .

, without the arrow, indicates the time derivative of the column matrix v. Since a particular frame must be defined as part of the definition of v, and each element of v is a scalar, from a mathematical standpoint this time derivative is uniquely defined, i.e., it can only have one meaning.

If v might best be considered as simply a mathematical entity. In particular, if v cannot be integrated to get v. That is what is meant by the phrase "v is not holonomic." For an excellent discussion of this issue, the reader is referred to Appendix B of the textbook by Hughes [5].

Hughes uses the following notation for the cross product matrix:

(1.4)

Using the Hughes system, the dot product and cross product are translated into matrix form as follows:

(1.5)

(1.6)

is a matrix, such operations are not as easily defined.

Matrix operations

With due attention to the order of operations, the dot product and the cross product can be interchanged;

(1.7)

The matrix equivalent of this expression is

(1.8)

Note that parentheses are essential for the last expression.

Sometimes it is useful to take the derivative with respect to a column matrix. Consider the scalar product

(1.9)

The partial derivative of this expression with respect to v is

(1.10)

; this matrix equation is equivalent to the three scalar equations

(1.11)

One slightly more complicated case will be presented here. Suppose

(1.12)

where I is a 3x3 symmetric inertia matrix and T is the rotational kinetic energy. The derivative of T with respect to ω is

(1.13)

The most convincing way to verify this is to write the complete expression for the scalar T in terms of the elements of I and ω, and then take derivatives term by term.

Organization of this book

Chapter . Several forms of the mass matrix are derived, depending on factors such as whether an integrated or reduced body is defined for the FEM, whether the engine motion is included in the dynamics or prescribed externally, etc.

Chapter 3, Section 3.1 provides a brief description of how a sloshing wave in a fuel tank can be represented by a suitable mechanical analog, either as a point mass on a spring or a point mass on a pendulum. Section 3.2 provides a Newton-Euler derivation of the nonlinear forces (Coriolis and centrifugal) on a slosh mass. Section 3.3 discusses various issues that arise if the FEM contains hydrodynamic elements that model the effect of sloshing fuel.

Chapter 4 contains a nonlinear Newton-Euler analysis of a pendulum on a spherical joint. The resulting model can be used to represent either a pendulum model of sloshing fuel or a gimbaled engine. This model is of particular importance if engine deflections or sloshing wave amplitudes are large enough that nonlinear effects must be included in the simulation.

Chapter 5 provides a discussion of the forces and moments that go on the right-hand side (RHS) of the equations. These include effects such as aerodynamics as well as apparent forces that arise in an accelerating reference frame. The phenomenon of rigid-body jet damping, which arises due to flowing propellant, is treated in detail. This chapter ends with summary of how to compute the forces that go with each equation.

Chapter 6 discusses the important topic of engine interactions, or more precisely the coupling that may exist between the engine motions and the rest of the dynamics. Special attention is given to the topic of inertial and thrust vector coupling of gimbaled engines with bending, which gives rise to thrust vector servoelasticity (TVSE). Recommendations for how the engine actuators should be modeled in the FEM are also provided.

Chapter 7 shows how the equations of motion can be put into state-space form that is suitable for either time-domain or frequency-domain analysis. Linear perturbation methods are used to introduce approximations for effects such as follower forces and aeroelasticity, and their influence on linear system eigenvalues and frequency response is summarized.

Chapter 8 discusses the important issue of producing the inputs that are provided to a simulation. Established practice is to run a Monte Carlo analysis in which parameters such as thrust, flex frequency, etc. are given a dispersed set of values, rather than a single value. During a simulation, the FEM must change at periodic intervals as the rocket mass changes. Section 8.3 shows how to minimize the disruption that occurs in a simulation during these changes.

Chapter 9 introduces the topic of stabilization and control of flexible boost vehicles using feedback. Linear analysis techniques developed in Chapter 7 are applied to synthesize feedback control structures that provide the desired closed-loop response of the rigid-body dynamics. A model for a typical actuation system, a pressure-stabilized hydraulic thrust vector control actuator, is introduced.

Finally, Chapter 10 incorporates material presented in previous chapters and discusses practical considerations for the development of production computer simulations. A simple constraint method using Lagrange multipliers is shown to be adequate for the modeling of launch pad interfaces. Numerical integration concepts specific to the present application are discussed. The important topic of designing Monte Carlo simulations and assessing results using binomial and order statistics, particularly for flight certification, is introduced.

Bibliography

[1] A. Greensite, Analysis and Design of Space Vehicle Flight Control Systems, Volume I - Short Period Dynamics. [Tech. Rep. CR-820] NASA; 1967.

[2] M. Rheinfurth, Control-Feedback Stability Analysis. [Tech. Rep. DA-TR-2-60] U.S. Army Ballistic Missile Agency; 1960.

[3] D. Garner, Control Theory Handbook. [Tech. Rep. TM X-53036] NASA; Apr. 1964.

[4] J. Frosch, D. Vallely, Saturn AS-501/S-IC Flight Control System Design, J. Spacecr. 1967;4(8):1003–1009.

[5] H. Bauer, Vehicle Stability and Control. [Tech. Rep. SP-106, The Dynamic Behavior of Liquids in Moving Containers] NASA; 1966:225 Ch. 7.

[6] P. Hughes, Spacecraft Attitude Dynamics. Dover; 2004.

Chapter 2: The system mass matrix

Abstract

In this chapter, the fundamental dynamic equations of a flexible rocket with sloshing propellant and a gimbaled engine are derived from first principles. The detailed analysis of these features is applicable to many rocket configurations, but is particularly important for very large rockets. In the case of space launch vehicles, the motion of propellant sloshing within the fuel tanks is of great significance to the design as often more than 90% of the vehicle's liftoff mass is liquid propellant.

Keywords

finite element model; mass matrix; integrated body model; reduced body model; undeformed centerline; bending equation; slosh equation; engine equation

In this chapter, the fundamental dynamic equations of a flexible rocket with sloshing propellant and a gimbaled engine are derived from first principles. The detailed analysis of these features is applicable to many rocket configurations, but is particularly important for very large rockets. In the case of space launch vehicles, the motion of propellant sloshing within the fuel tanks is of great significance to the design as often more than 90% of the vehicle's liftoff mass is liquid propellant.

Sloshing propellant is usually modeled as a linearized pendulum or an equivalent spring, mass, and damper coupled to the vehicle rigid and elastic degrees of freedom such that the force and moment response of the mechanical analog matches that of test-correlated semi-empirical models of a rigid tank. The portion of the equivalent liquid mass that is not in motion is lumped into the rigid-body mass. The properties of the mechanical analog change as a function of propellant remaining and the vehicle acceleration.

Engine dynamics can also play a significant role in the global vehicle behavior. For very large booster systems, particularly space launch vehicles, the use of large thrust-vectored engines results in a total moving engine mass that is a significant fraction of the total vehicle mass. Engine position control is often provided by high-power hydraulic or electromechanical actuators. This combination of moving mass, high actuator loads, and the lightweight, flexible stage structure leads to a variety of coupling effects between the engines and vehicle that must be accounted for explicitly in the design.

In the following development, the equations of motion will be developed initially for a rocket with a single fuel tank and a single engine. Generalization of these techniques to the case of multiple tanks and engines is