On the Screw Axes and Other Special Lines Asso-ciated with Spatial Displacements of a Rigid Body

where and 5 are two integrating constants to be determined by initial conditions. The last term of equation Ml) Error = ^y -1 ) /3 sin \p Equation The required torque T3 for maintaining constant spin velocity is given by equation (38). IV. Conclusion The foregoing analysis of an offset unsymmetric gyroscope with oblique rotor has demonstrated the useful application of 3 X 3 matrices with dual-number elements for the description of the kinematic relations within a system involving a number of reference frames with no common origin. Results obtained thus far have led us to believe that the 3X3 dual transformation matrices, in view of their orthogonality properties and ease of adaption to tensor notation, will offer a meaningful alternative in the analytical treatment of the mechanics of a system of rigid bodies in spatial motion. It is anticipated that this mathematical technique will greatly facilitate the systematic investigation of the dynamics of a variety of spatial mechanisms. The resultant analytical expressions, coupled with recent advances in our fundamental knowledge of spatial kinematics [11], may bring a step closer to design application of these potentially useful mechanisms. V. Acknowledgment The author expresses appreciation for the support by National Science Foundation under Grant GK1250 which makes this work possible. VI. References DISCUSSION L. Maunder 2 Two earlier publications refer to related but different, configurations involving certain obliquities in the kinematic axes of a two-axis gyroscope in gimbals. The first 3 discusses the free motion of a symmetric rotor in gimbals in which the axes of rotation of the gimbal frames are precisely perpendicular to each other but the spin-axis of the rotor is misaligned with respect to the rotor's principal axis. The arrangement is analogous to a state of dynamic unbalance in the rotor, and t he solution describes a sustained vibration of the system at the frequency of spin, accompanied by a steady rate of drift. The second 4 also refers to the free motion of a symmetric rotor in gimbals. In this case, three misalignments are specified, one associated with nonperpendicularity of the two gimbal axes, and the other two with component misalignments of the rotor spinaxis, with respect to the corresponding principal axis of the inner gimbal, i.e., the direction of the spin-axis in an ideal configuration. The details of this analysis show that the misalignments cause small but complex changes in the mutational frequency of the system, in the phase separating the vibration of the two gimbals, and in the steady drift rate associated with vibration. The present paper is an interesting variant and should have useful application to the study of the effects of manufacturing and assembly errors on the performance of precision gyroscopes. R. Wenglarz 5 The /•. in the second part of equatio