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J. Peraire, S. Widnall16.07 DynamicsFall 2009Version 3.0Lecture L28 - 3D Rigid Body Dynamics: Equations of Motion;Euler’s Equations3D Rigid Body Dynamics: Euler’s EquationsWe now turn to the task of deriving the general equations of motion for a three-dimensional rigid body.These equations are referred to as Euler’s equations. The governing equations are those of conservation oflinear momentum L M v G and angular momentum, H [I]ω, where we have written the moment ofinertia in matrix form to remind us that in general the direction of the angular momentum is not in thedirection of the rotation vector ω. Conservation of linear momentum requiresL̇ F(1)Conservation of angular momentum, about a ﬁxed point O, requiresḢ 0 M(2)Ḣ G M G(3)or about the center of mass GIn our previous application of these equations, we speciﬁed the motion and used the equations to specifywhat moments would be required to produce the prescribed motion. In this more general formulation, weallow the body to execute free motions, possibly under the action of external moments. We consider thegeneral motion of a body about its center of mass, ﬁrst examining this in a inertial reference frame.1

At an instant of time, we can calculate the angular momentum of the body as H [I]ω. One possiblemethod to obtain the moments and the motion of the body is to perform our analysis in this inertialcoordinate system. We would of course align our coordinate system initially with the principal axes of thebody. We could then writeM G Ḣ G d/dt([I]ω) [I ]ω [I]ω̇(4)This would be a appropriate approach but the diﬃculty is keeping track of [I ] in the inertial coordinatesystem. The initial inertial axis, even if principal axis, will not remain principal axis, and the inertia ”seen”in this coordinate system will vary with time. So unless we are considering the motion of a sphere, forwhich all axis are principal and the inertia tensor is constant about all axis, we cannot get very far with thisapproach.Body-Fixed AxisWe formulate the governing equations of motion in an axis system ﬁxed to the body, paying the price forkeeping track of the motion of the body in order to have the inertia tensor remain independent of time inour reference frame. Given our earlier discussion of terms added to the description of motion in a rotatingand accelerating coordinate system, it may seem surprising that we can do this easily, but the statement ofconservation of angular moment about the center of mass, or about a ﬁxed point of instantaneous rotationholds if we include the changes in angular momentum arising from Coriolis Theorem. The general body isshown in the ﬁgure. We ﬁx the x, y, z axis to the body and instantaneously align them with x, y, z. Referring2

to the ﬁgure, we see the components of ω,—- ω1 , ω2 and ω3 —- and the components of the angular momentvector H, which in general is not aligned with the angular velocity vector. We also see the vectors ω2 j ω3 kapplied to the x axis, with corresponding components of ω sketched at the y and z axes. At this instantof time, the change in H will be due to actual time rate of change Ḣ plus the eﬀect of the instantaneousrotation of the x axis due to ω: the change in H due to the instantaneous rotation of the coordinate systemisḢ [I]ω̇ ω H(5)Equating the change in angular momentum to the external moments, we have the statement of conservationof angular momentum in body-ﬁxed axes.Mx Ḣx Hy ωz Hz ωy(6)My Ḣy Hz ωx Hx ωz(7)Mz Ḣz Hx ωy Hy ωx(8)This particular form of the equations of motion is valid for any set of body-ﬁxed axes. If the axes chosen areprincipal axes, then we may express the conservation of angular momentum in terms of moments of inertiaabout the principle axes,Mx Ixx ω̇x (Iyy Izz )ωy ωz(9)My Iyy ω̇y (Izz Ixx )ωz ωx(10)Mz Izz ω̇z (Ixx Iyy )ωx ωy(11)These equations are called Euler’s equations. They provide several serious challenges to obtaining the generalsolution for the motion of a three-dimensional rigid body. First, they are non-linear (containing products ofthe unknown ω’s). This means that elementary solutions cannot be combined to provide the solution for amore complex problem. But a more fundamental diﬃculty, is that we do not know the location of theaxis system, x, y and z. Recall that since the axes are ﬁxed to the body, we are committed to follow thebody as it rotates in order to use these equations to obtain a solution. Thus, we must develop a methodto follow the changes in axis location as the body rotates. Before turning to this problem, we examine asituation where we know the location of the axis, at least approximately.Stability of Free Motion about a Principal AxisConsider a body rotating about the z axis—-a principal axis—- with angular velocity ωz . Without loss ofgenerality, we may consider this to be a rectangular block. We take advantage of the fact that we know to3

a good approximation the axis of rotation, at least initially. We examine the question of stability to smallperturbations, rotations about the x and y axis of magnitude ωx and ωy where ωx ωz and ωy ωz .The moments of inertia about the x, y and z axes are Ixx , Iyy and Izz ; we say nothing about the magnitudesof these inertias at this point.Assume a small impulsive moment that initiates a small rotation about the x and y axes and thereafter themotion proceeds with no applied external moments. For this case, Euler’s equations become0 Ixx ω̇x (Iyy Izz )ωy ωz(12)0 Iyy ω̇y (Izz Ixx )ωz ωx(13)0 Izz ω̇z (Ixx Iyy )ωx ωy(14)The z equation contains the product of two small terms in contrast to the other two equations where the sizeof the terms is comparable–as far as we can tell. We therefore note that since Izz ω̇z (Ixx Iyy )ωx ωy withboth ωx and ωy being small quantities, we may take ωz as constant, equal to some ω. We now diﬀerentiateboth equations with time, and substitute to obtain an equation for ωx . (We could do this as well for ωy withthe same result.)Ixx ω̈x (Iyy (Izz )(Izz Ixx ) 2ω ωx 0Iyy(15)orω̈x Aωx 0(16)The solution to this diﬀerential equation for ωx (t) is ωx (t) BeAt Ce At(17)The stability of the motion is determined by the sign of A. If A is positive, an exponential divergencewill result, and the initial small perturbation in ωx will grow without bound, as least as predicted by small4

perturbation analysis. If the sign of A is negative, oscillatory motion of ωx (and ωy ) will result, and themotion is stable.Examining A (Iyy Izz )(Izz Ixx ),Iyywe see that the condition for A to be positive is that Izz is intermediatebetween Ixx and Iyy . That is Ixx Izz Iyy or Iyy Izz Ixx . We conclude that a body rotating aboutan axis where the moment of inertia is intermediate between the other two inertias, is unstable. Also, thatrotation about either the largest or smallest inertias is stable. This consideration relates to stability of arotating body as predicted from Euler’s equation; we have already examined the stability of rotation aboutthe smallest inertia axis and concluded that if energy is dissipated, that motion is unstable.Stability of a GyrostatThe analysis of the proceeding section can easily be extended to a more important, more useful and morecomplex conﬁguration: the gyrostat satellite. A gyrostat consists of a spinning body which contains withinitself another spinning body, referred to as the rotor. Gyrostat satellite are used when the external body ofthe satellite must spin slowly to accomplish its mission while it needs the stabilization provided by fasterrotation. This is accomplish by placing a rotor inside the satellite. The angular momentum of the rotor isdriven by a motor attached to the satellite; no net momentum increase occurs when an adjustment in rotorspeed is made. The examination of this device is quite straightforward if the rotor principal axes are alignedwith the satellite principal axes, the rotor is axisymmetric, and the center of mass of the rotor is placed atthe center of mass of the satellite. Thus the rotor is constrained to move with the rotational motion of thesatellite and principal axes remain principal.5

The moment of inertia of the rotor about its principal IR xx [I R ] 0 0axes is00RIxx00RIzzwhile the moment of inertia of the satellite platform is IP 0 xx P[I P ] 0Ixx 0000PIzz (18) (19)Because of the geometric constraints on the system, ωx and ωy are equal for both platform and rotor whileωz diﬀer for platform, ωzP and rotor, ωzR . Since we are in a rotating coordinate system with respect to theplatform so the Ixx and Iyy are constant, ωzR is the relative rotation velocity between platform and rotor.Therefore the total angular momentum is PIxx [H] 0 000PIyy00PIzz ωx ωy ωzP PIxx 0 000PIyy00PIzz ωx ωy ωzRBecause of the constraints on ω, we can write the angular momentum as PRIxx Ixx00ωP x PPR[H] 0 ωyIyy Ixx0 PR R00Izz Izzωz /ωzPωzP (20) (21)where ω has been identiﬁed as ω P .This is a remarkable result. It indicates that the eﬀect of the rotation on the angular momentum of thegyrostat can be incorporated by a modiﬁcation of the moment of inertia about the z axis. We will call thisthe ”eﬀective” moment of inertia,PR RIzz ef f Izz Izzωz /ωzP(22)where, ωzR is the relative rotation velocity between platform and rotor.With this result, and the identiﬁcation of the ”eﬀective” moment of inertia in z, the previous analysis of thestability of rotation of a body with unequal moments of inertia goes through with the replacement of Izz byIzz ef f .ADDITIONAL READINGJ.L. Meriam and L.G. Kraige, Engineering Mechanics, DYNAMICS, 5th Edition 7.9J. B. Marion, S. T. Thornton, Classical Dynamics of Particles and Systems, 11.8W.T. Thompson, Introduction to Space Dynamics, Chapter 5F. P. J. Rimrott, Introductory Attitude Dynamics, Chapter 116

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