linear vibration analysis using screw theory - helix - Georgia Institute ...

1123·~T M =¸ T 1 T 2 T 3=640 0 00 0 L=20 ¡L=2 01 0 00 1 00 0 175(2.15)where the subscript M indicates that the eigenwrenches and eigentwists are expressed at point M.The eigenvalue problem also yields,·~k f = diag·~k ° = diagAYLGJL12Y I zL 3Y I zL12Y I yL 3Y I yL¸¸(2.16)(2.17)A simpler expression for the eigenwrenches and eigentwists can be had by using a rigid body translation(2.5) and expressing them at the midpoint of the beam (E),2 3 2 3~w E =641075~T E =640175 (2.18)where 1 is the 3 £ 3 identity matrix. Comparing (2.2) and (2.18) reveals that the eigenwrenchesare all pure forces and intersect at the mid-point of the beam. Furthermore, the eigentwists areall pure rotations, they also intersect at the mid-point of the beam and they are collinear with theeigenwrenches. When a pure force eigenwrench is collinear with a pure rotation eigentwist, it isde…ned as a compliant axis. Hence a cantilever beam has three compliant axes as shown in Figure2.3.