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

159Finally substituting (7.2) leads to,k f1 w f = K¡w f (7.5)It is easily seen from (7.1) and (7.5) that the new wrench w f is also an eigenwrench of K. Hence,any linear combination of two eigenwrenches with equal linear sti¤nesses is also an eigenwrench. Asdiscussed previously, all the linear combination of two wrenches form a cylindroid. Hence, a sti¤nessmatrix having two equal linear sti¤nesses is characterized by a cylindroid of eigenwrenches. Thethird eigenwrench is parallel to the central axis of the cylindroid since it has to be orthogonal to allthe wrenches in the cylindroid as noted in Chapter 2.If the sti¤ness matrix has three equal linear sti¤nesses then by using the same proof as above,it easy to prove that any linear combination of the three eigenwrenches is also an eigenwrench. Forthis case, the eigenwrenches form a set of concentric hyperboloids with constant pitches.Obviously, the same properties are associated with the eigentwists when the rotational sti¤nessesare equal. In the following section, it is assumed that whenever the eigenwrenches form a cylindroid,the two wrenches that intersect at right angles are chosen as the eigenwrenches (there is only one pairof wrenches that meet this criteria in a cylindroid). Similarly, whenever the eigenwrenches form a setof hyperboloids, the three wrenches which intersect at right angles are chosen as the eigenwrenches(once again there is only one set of three eigenwrenches which meets this criteria). These resultsare used in the next section to examine the conditions on the eigenwrenches and eigentwists for thecenters of elasticity, sti¤ness and compliance to be coincident.7.2 Combined Elastic CentersAll the theorems in this section have been proven previously in [9] and [37], but a more insightfulproof is o¤ered here. The theorems give the conditions for the centers of elasticity (E) and sti¤ness