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  • NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

    Technical Report 32-1593

    Analytical Dynamics and Nonrigid

    Spacecraft Simulation

    P. W, Likins

    ANALYTIC&L DYNABICS ANDt_'7-'4- 313 3 3'!

    ,'(N_S&-CR-139502)'BONRIGID SPACECRAFT SIBUL&TIOB

    _Propulsion Lab.)

    (Jet

    SCL 22B Unclas

    G3/31 45755

    JET PROPULSION LABORATORY

    CALIFORNIA INSTITUTE OF TECHNOLOGY

    PASADENA, CALIFORNIA

    July 15, 1974

    ~

    ,_

    https://ntrs.nasa.gov/search.jsp?R=19740023220 2018-06-16T17:01:21+00:00Z

  • NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

    Technical Report 32-1593

    Analytica/ Dynamics and Nonrigid

    Spacecraft Simulation

    P. W. Likins

    JET PROPULSION LABORATORY

    CALIFORNIA INSTITUTE OF TECHNOLOGY

    PASADENA, CALIFORNIA

    July 15, 1974

    I

  • Prepared Under Contract No. NAS 7-100

    National Aeronautics and Space Administration

  • Preface

    The work described in this report was performed under the cognizanc_ of the

    Guidance and Control Division of the Jet Propulsion Laboratory, which is sup-

    ported by NASA contract NAS 7-100. The author is a Professor at UCLA and a

    com_ta.t to JPL.

    Precedingpageblank

    JPL TECHNICAL REPORT 32-1593 iii

  • Contents

    I. Introduction ...........

    A. Background and Motivation .....

    B. Scope of Study .........

    I1. Selected Methods of Analytical Dynamics .

    A.

    III

    , o ......

    Notational Conventions .....

    Definition of Symbols .....

    IV.

    References

    Appendix A.

    Appendix B.

    1

    1

    3

    4

    4

    4

    D'Alembert's Principle and Its Generalizations .....

    1. D'Alembert's principle ..............

    2. Lagrange's form of D'Alembert's principle for

    independent generalized coordinates ....... 5

    3. Lagrange's form of D'Alembert's principle for

    simply constrained systems ........ - . . 23

    4. Kane's quasi-coordinate formulation of D'Alembert's

    principle ................. 25

    B. Lagrange's Equations .............. 29

    1. Lagrange's equations for independent generalizedcoordinates ................ 29

    2. Lagrange's equations for simply constrained systems 43

    3. Lagrange's quasi-coordinate equations ........ 47

    C. Hamilton's Equations ............ 51

    1. Hamilton's equations for simply constrained systems 51

    2. Hamilton's equations for independent generalizedcoordinates ............... 55

    Application to Nonrigid Spacecraft .......... 57

    A. Multiple-Rigid-Body System Models ........ 57

    1. Single rigid body ........ 57

    2. Rigid body with simple nonholonomic constraints . 72

    3. Symmetric three-body system with small deformations 86

    4. Point-connected rigid bodies in a topological tree ..... 107

    B. Rigid-Elastic Body System Models ....... 119

    1. Single elastic body with small deformations ...... 119

    2. Interconnected rigid bodies and elastic bodies ..... 124

    Conclusions and Recommendations .......... 126

    130

    133

    136

    JPL TECHNICAL REPORT 32-1593

    PRiCkING PAGE BLANK NOT FILMED

    V

  • Figures

    1. Particle system constrained as rigid body ....... 9

    2. System of seven particles and two multiple-particlerigid bodies ................ 10

    3. Description of a deformable body ....... 11

    4. An example illustrating two floating reference frames .... 20

    5. Sphere rolling without slip ........... 73

    6. Attitude angles for the rolling sphere ....... 74

    7. Symmetric three-body system ........... 87

    vi JPL TECHNICALREPORT32-1593

  • Abstract

    This report contains an exposition of several alternative methods of analytical

    dynamics, and the application of these methods to alternative models of nonrigid

    spacecraft. This information permits the comparative evaluation of these methods

    for spacecraft simulation.

    The following methods are developed from D'Alembert's principle in vectorform:

    (1) Lagrange's form of D'Alembert's principle for independent generalizedcoordinates.

    (9.) Lagrange's form of D'Alembert's principle for simply constrained systems.

    (8) Kane's quasi-coordinate formulation of D'Alembert's principle.

    (4) Lagrange's equations for independent generalized coordinates.

    (5) Lagrange's equations for simply constrained systems.

    (6) Lagrangian quasi-coordinate equations (or the Boltzmann-Hamel equations).

    (7) Hamilton's equations for simply constrained systems.

    (8) Hamilton's equations for independent generalized coordinates.

    Applications to idealized spacecraft are considered both for multiple-rigid-body

    models and for models consisting of combinations of rigid bodies and elastic bodies,

    with the elastic bodies being defined either as continua, as finite-element systems,

    or as a collection of given modal data. Several specific examples are developed in

    detail by alternative methods of analytical mechanics, and results are comparedto a Newton-Euler formulation.

    Conclusions are straightforward in the case of the multiple-rigid-body topo-

    logical tree idealization, for which the standard of comparison is a Newton-Euler

    formulation due originally to Hooker and Margulies and widely available in the

    form of a JPL computer program.

    Although the equations in the previously existing JPL computer program are

    obtained in this report by means of both Kane's approach and the Lagrangian

    quasi-coordinate method, neither these nor any other methods of analytical dy-

    namics produced results superior to the present standard.

    Applications to combinations of rigid bodies and elastic bodies are more varied

    and more complex, and conclusions are more tentative, but essentially the same

    result emerges. Although various methods of analytical dynamics produce the

    same equations of motion as have previously been derived by the Newton-Eulerapproach, there appears to be no demonstrable advantage in any of the methods

    of analytical dynamics over the Newton-Euler results, except in the unusual case

    in which a continuum idealization is appropriate and in the somewhat academic

    case in which a truncated set of vibration mode shapes and frequencies are given

    in advance of the dynamic analysis.

    JPL TECHNICAL REPORT 32-1593 vii

  • Analytical Dynamics and NonrigidSpacecraft Simulation

    I. Introduction

    A. Background and Motivation

    In the traditional academic perspective, the classical methods of Lagrange and

    Hamilton are, in comparison with the direct application of Newton's laws,

    accepted as the more advanced procedures for formulating equations of motion

    for mechanical systems.

    The methods of Newton and Euler, which involve physically visualizable quanti-

    ties represented in modern times by Gibbsian vectors _ and dyadics, are generally

    recognized as being most useful in the struggle for conceptual understanding of

    the behavior of relatively simple systems, such as particles in space or gyroscopes.

    It is, however, widely believed that, in providing the transition from the physical

    world of vectorial mechanics to the abstract analytical realm of generalized scalar

    formulations found in analytical mechanics, Lagrange gave us superior procedures

    for deriving equations of motion for complex mechanical systems.

    Hamilton's formulations are widely regarded as even more powerful than those

    of Lagrange. Hamilton's principle embraces much of Newtonian mechanics in a

    single, scalar variational equation; Hamilton's canonical equations replace

    Lagrange's scalar, second order, ordinary differential equations with first order ordi-

    nary differential equations of remarkably simple structure; and the Hamilton-Jacobi

    equation is a single partial differential equation that subsumes much of Newtonian

    and Lagrangian mechanics.

    XAGibbsian vector (to be distinguished from an n-dimensional column matrix) is geometricallyequivalent to a directed line segment in physical space, with rules for addition and both scalarand vectorial multiplication.

    JPL TECHNICAL REPORT 32-1593 1

  • The methods of Lagrange and Hamilton automatically remove from the equa-tions of motion most of the unknown and unwanted forces of constraint that

    plague the analyst who applies Newton's laws. Moreover, the former methods

    yield differential equations whose structure is system-invariant, while the proce-dures of Newton and Euler must be reconstructed for each new mechanical

    system. Finally, the equations of Lagrange and Hamilton are explicitly constructed

    to facilitate integration, whereas those of Newton and Euler have no particular

    structure at all, being dependent for their form on the strategy adopted by the

    analyst.

    Against this background, we consider the notoriously complex problems of

    formulating equations of motion for nonrigid spacecraft. Probably no other class

    of physical system is routinely subjected to such complicated mathematical

    modeling, and described by such difficult ordinary differential equations of motion.

    Fortunately, the spacecraft and its physical environment are much more amenable

    to accurate modeling than are other physical systems of comparable or greater

    complexity (such as the automobile, or the human being). The internal structure

    of the spacecraft is subject to component testing and design control, and the

    external space environment is much less complex than the terrestrial