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[[?]] = S[[subscript]]n[/subscript]](1)
[[?]] = dimensionless cantilever mode shape
η = dimensionless coordinate along the beam axis
Ω = natural frequency.

Taking advantage of the fast convergence of the 'cubic' mode shapes (which is similar to the assumed elastic deformation distribution function in simple finite beam elements), the following Sn is used in ASAD:

Sn=Cn7^2 (1+dn7^2)
15d^2n + 24 dn+7=0
Cn= (105/15d^2n+35dn+21)^1/2}(7)

Eliminating a, ω, T and F, the system motion equation can be shown to be:

[[??]] (8)

where [[?]] = [[?]]

[[?]] = Orbiter angular acceleration,

[[?]] = angular accelerations of the joints about their drive axes,

[[?]] , [[?]] = angular accelerations of the joints about two cross axes, normal to the drive axes,

[[?]] = beam torsional accelerations,

[[?]], [[?]] = beam bending accelerations in generalized coordinates,

[[?]] = system inertia matrix which is usually full,

[[?]] = system stiffness matrix which has zero rows and columns corresponding to [[underline]] Ø [[/underline]] and [[underline]] Y [[/underline]], banded diagonal submatrices corresponding to [[underline]]α[[/underline]] and [[underline]] ß [[/underline]], and non-zero diagonal submatrices corresponding to [[underline]]Θ[[/underline]], [[underline]]p[[/underline]],and [[underline]]q[[/underline]],

[[underline]]t[[/underline]] = applied torque + damping torque + nonlinear inertial torque. The applied torque consists of the torque on the Orbiter due to the Orbiter Reaction Control System jet firing and the torques delivered by the SRMS gearboxes.

The 'static' arm frequencies and mode shapes are computed from the submatrices of [[??]] and [[??]] corresponding to the non-zero submatrix of [[??]]. These frequencies and mode shapes are compared with the finite-element model results as a means to validate ASAD. They are also used to transform (8) into an equation in 'arm mode' generalized coordinates. The transformed equation is partitioned into two equations in terms of the (low frequency) active modal coordinates and the (high-frequency) passive model coordinates. The active modal coordinate equations are integrated numerically while the passive modal coordinate equations are treated quasi-statically by ignoring the acceleration and velocity terms. The integrated active and computed passive coordinates are finally transformed into physical coordinates for display purposes. The above modal reduction method is used simply for economy reasons while maintaining the simulation accuracy.

Figures 16 and 17 show a sample payload motion simulated by ASAD. The arm is loaded with the 14615 Kg payload which is commanded to move along the Z axis of the Orbiter in the Operator Commanded Automatic mode. After approximately 105 seconds, the automatic sequence is terminated, and the joint servos are automatically switched to Position Hold mode to maintain the arm configuration which exists at the end of the automatic sequence. The X,Y, Pitch, Yaw and Roll motions are uncommanded motions arising from arm flexibility and different servo transient responses. 

[[3 figures]]
Figure 16 Translational Motion of the Point of Resolution in Response to Z Command in Automatic Mode]]
[[/3 figures]]

 

Transcription Notes:
Some symbols and equations I was unable to generate accurately on my keyboard, so I used their closest counterparts. (in response to above) Should the symbols be described? for example: S sub n Should the equations be included (like equation 7)?