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

where ~r S f and ~rC °are respectively the perpendicular vectors from the center of sti¤ness to the eigenwrenchesand from the center of compliance to the eigentwists. Equation (2.33) shows that each setof vectors are coplanar respectively at S and C.The location of the centers of elasticity, sti¤ness and compliance are important parametersa¤ecting the modal response of an elastically suspended rigid body. For planar motion, the threecenters coalesce into a unique point on the plane and as shown above, the sti¤ness matrix becomesdiagonal when expressed at this point. The diagonal elements of the sti¤ness matrix are then the16principal sti¤nesses similar to the principal inertias for the mass matrix.This unique point isalso referred to as the point where there is no dynamic coupling in the equation of motion[43]. Indistinction to the planar case, the spatial 6 £ 6 sti¤ness matrix cannot be generally diagonalized bya rigid body transformation[31]. Although, in some cases like the beam example presented above,the sti¤ness matrix is diagonal at the center of elasticity which in this example is at the midpointof the beam.2.4 Decomposition of the Mass MatrixThe mass matrix can be used to model the inertia of more than one rigid body. For example, itcan be used to model so-called articulated inertias (see [20]) in which case, the mass matrix becomesa full symmetric 6 £ 6 matrix with 21 independent parameters. Articulated inertias are very usefulin analyzing the dynamics of a robot.Lipkin[30] proposed a decomposition for the mass matrix which is similar to the one proposedfor the sti¤ness matrix. His decomposition is also based on two eigenvalue problems. Consideringonly the inertial response of the body shown in Figure 1.1 which is initially at rest, the eigenwrenchproblem becomes, what wrenches (u f ) can be applied to the body so that the resulting accelerationtwists are pure translational accelerations parallel to the wrenches? The eigentwist problem is stated