Thus, given an observed variation in the spin period and the average which may be substituted for /? in Eq. (16-124), we can compute the nutation angle, 8.

Equation (16-124) may be derived by extending the development of Flatley [1972a]. Let the inertial frame be as before with the Z axis collinear with L and the X axis along die projection of S into the plane perpendicular to z. Let the Sun sensor slit contain the body x and z axes. Then, Eq. (12-20) shows that the j> component of S in the body frame, assuming 0 small, is

Sf = sin P( - sin if* cos $ - cos xfi cos 0 sin $)+cos P cosf sin 0

and is equal to zero at a Sun sighting. At any time, /, Eq. (16-100) gives for a single-spin symmetric spacecraft

/jiOStf

The first Sun sighting after 1=0 will occur at a rotation angle of approximately 2«-(4>0+ V>o)modi«: general, the nth Sun sighting will occur at

where Sla is the deviation of the crossing time from that if 0=0.

Using Eqs. (16-126) and (16-127), and defining = Eq (16-125) becomes

(PtOor 2tt)

Thus, the period between any two consecutive Sun sightings is

Note that

assuming R^iHt^-&„_i) is small. Substituting Eq. (16-132) into Eq. (16-131) and reducing yields

To find the minimum and maximum P„, we note that only the factor sin(i>„ + nR,) varies. The extrema of the sine function are ± I and occur at

^n + vR1 = v/2±iri i=0,1,2,3 for p 9» 0 or v, a, ?» 0, and R, 9» 0,1,2,3, Furthermore,

(max for <p„+ vRt< min for if>„ + vRj =

for 0<p<v/2 and 2/'<rt,<2/"+l. For v/2<p<v or 2i-\<R,<2i, the maximum and minimum values for P„ are reversed. Substituting Eqs. (16-133) and (16-135) into Eq. (16-123) yields Eq. (16-124).

For the case where R, approaches an integer, Eq. (16-133) shows that Pn approaches In/a, for all n; i.e., the spin rate variation becomes small with respect to the other approximations. The same phenomenon occurs as (i approaches v/1. For either of these cases, simulations are necessary to determine the spin rate variation arising from the second-order effects neglected above.

The phase, of L in the spacecraft coordinate system at the time of a maximum spin period measurement is determined from Eq. (16-134) and is summarized as follows:

Note that $ is measured clockwise from the +y axis so that in terms of the azimuth, £ (measured counterclockwise from the Sun sensor slit plane or the x axis) a maximum occurs at either ^ = 270°-180°*, = 90°-£ or at £= -180°+ 180°*,.

The variation in the Sun angle for a symmetric spacecraft may also be used to determine the phase of L in the body coordinate system. Applying Eq. (12-20) to the Sun vector, S, gives the z component of S in the body frame as

«0sin/8sin$+cos0cos0 (16-136)

Substituting from Eq. (16-126) for 4> and from Eq. (16-128) for t at a Sun sighting, we have sin^=sin[(*,«xf„- ^o+^o+^f,]

where Eq. (16-127) has been used to identify — ip. Thus, cos0ms»-0sin/3sinii< + cos0cos0 (16-138)

which shows that pm varies with the period of rp, and is a maximum when \(/ = 90 deg (£=0) and a minimum when \p=270 deg (|= 180 deg) for a slit in the x-z plane. This verifies the statement made earlier that the period of oscillation of the measured Sun angles for a symmetric spacecraft which measures the Sun angle only once per spin period is that of if>.

The spin period variation observed from an asymmetric satellite depends on the orientation of the slit relative to the x and jr axes. A convenient approximation based on a number of simulations with a dynamics simulator (ADSIM, described by Gray, et al„ [1973] is

where AP is defined by Eq. (16-123), is the maximum nutation angle in degrees, and R, is defined by Eq. (16-121). Rp has the approximate analytic form

and can be obtained more accurately either from Fig. 16-9 or by solving Eq. (16-118).

Note that R£ has the opposite dependence on | from that observed with the Sun angle variation. (See Eq. (16-119).) That is, the maximum spin period variation occurs for a sensor on the x axis (I, < Iy), whereas die maximum variation in Sun angle occurs for a sensor on the ^ axis. Simulations have shown that Eq. (16-139) holds for 0< 10 deg and but that the spin rate variation does not go to zero as die Sun angle approaches 90 deg as Eq. (16-139) indicates. For a specific spacecraft, simulations are recommended to obtain a more accurate relationship between AP and 0. Similarly, simulations are necessary to determine the orientation of L at the measurement of the maximum spin period because of the complex relationship between the variations of \f> and the geometrical effects observed in the Sun angle variation.

To illustrate the usefulness of the spin period variation, consider the spacecraft in the previous example. For those moments of inertia Rj = 0.35, R, = 0.442, and (from Fig. 16-9 or Eq. (16-140)) ^¿ = 0.915 for £ = 30 deg. Assume j3ra = 50 deg and the observed spin period ranges between 6.0288 sec and 5.9707 sec; then g

AP=4.84X 10~3= cot(50° )sin(79.56° )0.915 or 0,^ = 1.15 deg.

16.4 Flexible Spacecraft Dynamics

Flexible body dynamics becomes significant when the natural frequencies of flexible spacecraft components have the same magnitude as spacecraft rigid body frequencies due. to either librational motion of a gravity-gradient stabilized spacecraft (Section 18.3), nutation of a spin-stabilized spacecraft (Section 16.3), or control system response of an actively controlled spacecraft. (Section 18.3). The lowest natural frequencies of flexible components should be at least an order of magnitude greater than the rigid body frequencies'before flexibility can be safely neglected.

The uncoupled lowest natural frequency, /, of a typical experiment boom with an end mass, M, extending from a compact, nonspinning central body can be estimated by the following equation, derived from linear beam theory:

where £ is the Young's modulus of the boom structural material, / is the area moment of inertia of the boom cross section, p is the boom mass density per unit length, and / is die boom length. The product EI is the bending stiffness, which can be computed or obtained from, experimental results. Typical values range from 6.0

N-m2 for a very flexible antenna element such as those on the RAE, to 170 N-m2 for stiff spin axes booms on spinning spacecraft.

When the estimated natural frequencies of flexible components are close to the rigid body frequencies, a more detailed analysis of flexibility effects is warranted. Deformations in very flexible spacecraft will strongly influence the magnitude and distribution of external and internal forces. Because the internal dynamics can be highly nonlinear, a rigorous time history simulation of the spacecraft system is required to predict attitude motions.

The need for simulation of flexible spacecraft dynamics depends on the attitude determination accuracy required because all spacecraft are flexible at some level. For attitude data, flexibility effects will be exhibited as either superpositions of a high-frequency signal or as the dominant portion of the attitude motion, depending on the flexibility of the system. Therefore, it is important to compare the effects of flexure with the attitude requirements and with sources of error other than flexibility. Attitude determination errors can be present in highly flexible spacecraft because of the relative motion between the attitude sensors and experiments that require precise attitude measurements. Such systems may require additional sensors to determine the position of the experiment relative to the prime sensor.

Flexible components interact with the spacecraft attitude control system by superimposing deflections and accelerations on the average measurements made by attitude sensors and rate gyros. Consequently, the control system can give erroneous command signals that could destabilize a spacecraft. Flexibility can also move the instantaneous center of mass and moments of inertia and thereby induce unexpected responses to command control torques.

16.4.1 Flexibility Effects on Spacecraft Attitude Dynamics

A quantitative analysis of flexible spacecraft attitude dynamics is beyond the scope of this section. (See, for example, the conference proceedings edited by Meirovitch [1977]). However, we will discuss specific effects in general terms to make the reader aware of the various phenomena that may occur. A particularly good review and bibliography is presented by Modi [1974].

Gravity-Gradient Forces. Gravity-gradient forces are both space- and time-dependent when acting on long flexible components such as the RAE antenna booms. When large, the deformations cannot be treated by simple linear methods because of the change in loading as the boom deforms. Time-dependent loading is induced by libration of the spacecraft resulting from orbital eccentricity. Large deformations will change the principal moments of inertia of the spacecraft system and influence the observed attitude motions. However, axial tension due to gravity-gradient forces can significantly increase the effective bending stiffness, thereby raising the natural frequencies.

Solar Heating. Temperature gradients due to unequal solar heating can cause warping of spacecraft structures. The effect on the spacecraft attitude depends on the time history of the solar energy input, structural properties (including cross-

section geometry), thermal expansion coefficients, thermal conductivity and diffu-sivity, and surface properties (absorptivity and emissivity). In a fully sunlit orbit, solar heating on an Earth-pointing spacecraft can cause a bias in the attitude or induce attitude motion with a frequency equal to the orbit rate. As a satellite passes in and out of the Earth's shadow, transients can be induced by the step changes in thermal loading. The significance of these transient loadings will depend on whether the flexible spacecraft has natural frequencies that are close to multiples of the orbital frequency.

Some early spacecraft (Naval Research Laboratory's gravity-gradient satellite 164 [Goldman, 1974] and OGO IV and V [Frisch, 1969]) experienced stability problems that were attributed to solar thermal deformations. These problems have generally been overcome by designing deployable elements for minimum temperature gradients and increasing their torsional stiffness. Thermal effects can still be important, however, if the attitude is critical for experiment sensors on the end of a long boom. The attitude change, 9, at the end of a boom due to a temperature gradient can be approximated by where a is the coefficient of thermal expansion of the boom material, AT is the temperature difference across the boom, d is the boom diameter, and / is the boom length. Typically, a is of order of 1.8x 10-5 cm/(cm°K) and A7" is in the range 0.3°K < AT < 0.8°K.

Temperature gradients in spinning spacecraft are generally not important due to the averaging effect of the spin rate. However, perturbations due to thermal lag could develop when experiment booms have a long thermal time constant and are shadowed by the spacecraft body once per spin period.

Deployment Dynamics. Coriolis forces are developed during boom development as a result of the deployed components moving relative to the body axes with a deployment velocity, v, and the body axes themselves rotating at an angular rate S2 with respect to an inertial frame. The Coriolis acceleration, 2S2 X v, during deployment reduces the spin rate and deforms flexible appendages in a direction opposite the direction of rotation. When deployment stops, the restoring forces due to strain, centrifugal, and gravity-gradient forces will cause the flexible elements to oscillate in phase about an equilibrium position. A periodic motion will therefore be superimposed on the spin rate. The persistence of this motion will depend on the effectiveness of structural damping or boom damper devices.

Solar Pressure. Solar torques (Section 17.2) are modified in flexible spacecraft by the change in the instantaneous angle of incidence of the solar radiation due to deformations. The differential force acting on a mass element of a flexible member is proportional to the cosine of the local instantaneous angle of incidence. For very flexible spacecraft, the dynamical system is nonlinear, because the loading becomes a function of the deformation. In addition, spacecraft deformations can induce solar pressure torques due to the shift of the center of pressure from the center of mass. For most satellites at low or intermediate altitudes (up to 6500 km, solar torques due to spacecraft flexibility are negligible when compared with gravity-gradient torques. At synchronous altitude, however, solar pressure can have a significant impact on the stability of gravity-gradient stabilized spacecraft.

Aerodynamic Drag. Deformations of flexible spacecraft modify rigid body aerodynamic torques in a manner analogous to solar pressure torques. Again, the magnitude of a differential force acting on a mass point is a function of the instantaneous angle of incidence of the air stream. Below 500 km, aerodynamic drag forces can induce significant deformation on highly flexible Earth-pointing spacecraft. The shift of the center of pressure from the center of mass due to the deformation may induce destabilizing torques that could tumble nonspinning spacecraft. Spinning spacecraft with transverse wire booms will tolerate high aerodynamic pressures if spin rates are at least 5 rpm. Aerodynamic forces will deform the wire booms; however, simulations have demonstrated that the energy absorbed during half a revolution is removed during the other half of the revolution. Spinning spacecraft in low-perigee orbits will exhibit boom oscillations but insignificant attitude perturbations due to aerodynamic drag.

System Frequencies and Modes. Spacecraft with more than one flexible boom have system frequencies that depend on the phase relationship of boom displacements with respect to each other. The system frequencies and modes are not the same as structural bending modes because they are a combination of all flexible element modes. Antisymmetric modes will induce rotation of the central spacecraft body that will be detected by attitude sensors, as shown in Fig. 16-12. Symmetric modes do not couple to attitude motion. Hence, large symmetric element deformations cannot be sensed by attitude sensors alone. Additional position sensors are necessary when information concerning the deformed shape of an experiment boom is critical to its performance.

EQUILIBRIUM STATE SYMMETRIC DEFORMATION ANTISYMMETRIC DEFORMATION

Fig. 16-12. System Modes of a Gravity-Gradient Stabilized Spacecraft With Flexible Booms. ^ is the pitch angle.

Attitude Perturbations Due to Thrusting. The dynamic response of a flexible spacecraft to thrusting can result in undesirable perturbations of the spacecraft attitude. For example, the location of the center of mass, within the spacecraft may be time dependent due to deformational motion. Accordingly, the thrust from body-fixed nozzles used for orbit adjustment will induce both rotational and translational motion because the thrust vector will not pass through the instantaneous center of mass at all times. The rotational motion perturbs the attitude and changes the direction of the thrust vector.

Thrusting for attitude adjustment will cause deformation of transverse booms out of the spin plane. Repeated pulses can cause a buildup of deformations depending on the phasing of the pulses and deformations and may result in large attitude motions about the nominal rigid body orientation.

The complexity of the flexible spacecraft equations of motion is increased by the additional degrees of freedom required for structural deformations and the coupling between translational, rotational, and deformational motion. Several methods for derivation of the equations of motion for computer simulation of flexible spacecraft are given by Likins [1970]. The generalized system mass matrix formulation is presented to illustrate the type of system equations encountered in the simulation of a flexible spacecraft.

These equations are appropriate for a rigid spacecraft with flexible solar panels and antennas and, possibly, a set of momentum wheels within the rigid structure, as shown in Fig. 16-13.

Fig. 16-13. Rigid Spacecraft with Flexible Solar Panels and Antenna and Momentum Wheels Within

To specify the spacecraft structure and motion, we define the following quantities: the mass, m, and the moment of inertia, I, of the complete spacecraft in an equilibrium configuration; the location, r, and velocity, v, of the center of mass of the complete flexible spacecraft in its rigid frame of reference; the angular rate, <o, of the rigid spacecraft and the four-component attitude quaternion, q, defined in Section 12.1; the n-vectors

Fig. 16-13. Rigid Spacecraft with Flexible Solar Panels and Antenna and Momentum Wheels Within

'ANTENNA

the Rigid Structure

'ANTENNA

the Rigid Structure

which provide the modal coordinates and velocities of the flexible spacecraft

[Goldstein, 1950]. Here, n is the number of generalized coordinates which describe the small oscillations of the spacecraft due to its flexibility. In addition, we introduce the component vectors

where F is the total external force and N is the total external torque acting on the spacecraft of total angular momentum LIO). There may also be generalized forces driving the normal modes, which will be represented by the «-vector P2. Finally, we introduce the nXn diagonal damping matrix, Cu\ the «X« diagonal stiffness matrix, K22; and the (n + 6)X(n + 6) system mass matrix, M, which is formed from four matrices, as follows:

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