ROTATING UNCONSTRAINED MODES: A MORE APPROPRIATE DYNAMIC ANALYSIS OF FLEXIBLE SPINNING SPACECRAFT H. B. Hablani* Johnson Space Center Houston, Texas Abstract The objective of the paper i s a formation of ve- hicle modes for gyroscopic space structures through a continuum formulation. This requires examination of adjointness of the associated eigenvalue problems. It i s shown that the stand- ard conditions of self-adjointness are inadequate even for nongyroscopic structures; modificati ns required depend on the oroblen under study 8n one hand, i n the nongyroscopic case the frequencies appear only quadratically and it is relatively easy to devise the corresponding vehicle modes and the self-adjoint operators. On the other hand i n the case of the gyroscopic structures the frequencies appear quadratically as well as linear- ly; this takes the vehicle mode from a configura- tion space to a phase space, that is,a Hilbert space i n the present case. Four elastic systems of increasing complexity typifying nonspinning and spinning elastic spacecraft are investigated in detail. 1. Introduction Model ing errors, i n general , and truncation errors, i n particular, have been repeatedly im- puted to endanger our success i n controlling the future large space structures.] These errors are a1 1 the more enervating in the case of spacecraft which spin as a whole or i n part or which have stored angular momentum; this i s because the asso- ciated vehicle modes have never been defined. The available discrete methods for gyroscopic systems are not adequate to minimize the truncation errors; a precise formulation of the associated vehicle modes, from a continuum analysis, i s desired in addition, because we be1 ieve that these modes wi 11 illuminate the physical nature of the vibrations of the spinning spacecraft and will help construct reduced order models of high fidelity for control studies. Formulation of such modes i s the aim of this paper. For a clear understanding of our contrib tions .! we define the following hierarchy of modes. (la) Simple Constrained Modes: In this situation the rigid portion of the nonspinning space- craft i s constrained to be imnobile and the elastic portions are allowed to vibrate. The word 'simple' is prefixed above to re- mind that only elastic restoring forces and the inertial forces are acting on the structure. More common phrases such as 'canti lever modes, I 'fixed-base modes1 and I appendage modes ' mean t h e same. (lb) Simple Unconstrained Modes: In this case the entire ncnspinning vehicle vibrates and various parts of the vehicle interact dyna- mically and freely. ~u~hes.3 has theorized these modes to a great depth. For the rotating spacecraft we analogously define: (2a) Rotating Constrained Modes: A mode i n which the spacecraft is a1 lowed to spin ~lniformly *NRC/NASA Research Associate, Structures and , echani cs Division, Member AIAA This paper is declared a work of the U.S. Government and therefore is in the public domain. about the intended principal axis,and the attached spinning el asti c surfaces vibrate without affecting the uniform spin and the attitude of the rigid portions. This mode also may be called a 'uniform spin mode.' (2b) Rotating Unconstrained Modes: Like (lb) this is a vehicle mode. It allows the rigid body and the appendages to spin freely i n space; the dynamics of the rigid portion and the elastic structure are permitted to interact freely. Modes similar to (Za), (2b) can be defined for a spacecraft with stored angular momentum. The modes &la) are generally used by analysts see Likins ; the use of modes (lb), more accurate than the (la), to describe the dynamics of an elastic spacecraft is, increasing, Garg5 and ~oelaert6 are two examples. The modes ( l b ) , unlike the modes (la), afford precise metrics of motional inter- action between different parts of the spacecraft; these metrics play a foundational role in control studies and an extensive use of them can be re- viewed i n Ref. 7. For a greater modeling accuracy it i s desirable to analyze the dynamics of flexible spinning space- craft by using the modes (Zb), (Za), i n this order of preference, instead of the modes (la). An example of the use of the modes (Za), while deter- mining the stability of a flexible spinning satel- l i t e , can be seen i n Hughes and Fung8; however, the modes (2b) have never been formulated. The reason is, of course, that such a formulation re- quires difficult mathematical analysis. Construct- ing the modes (2b) for a specific spacecraft is our present objective. This paper presents a modal analysis of four idealized elastic systems of increasing complexity. A common feature of the four systems i s that each consists of a rigid body 67 having mass and moment of inertia and an elastic beam f , a study of the dynamic interaction between RandL is desired in all tne systems. In a differential fornulation of the vibratinns of such systems, boundary ccnditions involve eipenvalues; this complicates the examina- tion of self-adjointness or ortho~onality of the nodes. I n Sec. 2 de dill review, and show the in- adequacy of the text-book conditions of self-ad- jointness for the above dynainic systems; model (1) consistiny of a cantilevered with aneat the freo end i s used as an cxample. b!e further illustrdte that dependinn on the comnlexity of tbe eig~vaic? woblem i n haiid, one has to ing2nious I aed7s.o h icalar product to obtain the orthogomf :.rdes i, a function space. Sec. 3 i s concerned lwitn th2 un- o strained modes of a nonspinning system model 725': a freeru~th anaat the center, simu13tiny a soacecraft i n deep space. Several ,flathemati cal n&els are constrxted and h e i r self-adjointness is jnyestinated. The model 2 is a zero-spin limi tinq case of the model I 3 1 and the niodo; (4) analyzed i n Sec. 4. The .node? (3) spins .about the transverse ,wi'icipal axis, t$e deformations gf f and the attitude of&in the spin plane only arz considered; this ir,iplies nodeling the centripetat force field