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

32Theorem 10 When the center of mass does not lie on an axis of principal sti¤ness, then the threevibration centers must lie in the shaded area of Figure 3.2Theorem 11 When the center of mass lies on a principal sti¤ness axis, then the two …nite vibrationcenters also lie on that axis, one on each side of the centers-of-elasticity and mass.3.3.2 Orthogonality ConditionsThe orthogonality conditions also provide geometric constraints on the location of the vibrationcenters. Consider the orthogonality conditions for the mass matrix associated with the eigenvalueproblem written at the center of mass.³^T T M´i M M³^T M´j= 0 i 6= j i; j = 1; 2; 3 (3.22)where³^T M´= (X MV ) i ^T V and i, j denote the mode number. Rewriting (3.22) for modes i, ki(k 6= j) and subtracting from (3.22) results in,¡¡!MV i ¢ V ¡¡¡!k V j = 0 i 6= j 6= k i; j; k = 1; 2; 3 (3.23)where (3.3), (3.4), m x = m y were used and ¡¡¡! V k V j = ¡¡! MV j ¡ ¡¡! MV k is the vector from vibration centerV j to vibration center V k . Equation (3.23) reveals that the vector from the center of mass M to thevibration center V i is perpendicular to the vector connecting the remaining two vibration centers.Furthermore, consider the triangle formed by V 1 , V 2 and V 3 . If the three lines perpendicular to thesides of the triangle and through the opposing vertex are drawn, then these lines must intersect atthe center of mass (see Figure 3.5). It is also worth noting that (3.23) can also be obtained from(3.6)-(3.8). Hence, the following theorem has been proven.Theorem 12 The center of mass is located at the orthocenter of the triangle formed by the threevibration centers.