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

30yVMExFigure 3.2: Location of Vibration CentersTheorem 8 Translation modes are parallel to a principal sti¤ness axis with respective natural frequenciesof ! x and ! y .Theorem 9 For one translation mode, the two …nite vibration centers lie on the parallel principalsti¤ness axis. For two translation modes, the one …nite vibration center lies on both principal sti¤nessaxes, i.e. at the center of elasticity E.3.3 Geometric Interpretation3.3.1 Location of Vibration CentersFor a given system, the potential vibration center locations are not arbitrary but con…ned todistinct regions. The preceding analysis is used to identify these regions that can be speci…ed verysimply in terms of the center of mass, the center of elasticity, and the principal sti¤ness directions.Again the three basic cases are considered.Case 1, three distinct centers. Examining (3.6) and (3.7) shows that EV y and MV y must eitherbe both positive or negative and the same for EV x and MV x . The result is that the vibration centerscan only lie in the shaded portion of Figure 3.2.Case 2a, EM y = 0 (or EM x = 0). The two …nite vibration centers lie on the x-axis (y-axis).However from (3.7) ((3.6)) they cannot lie between E and M. Their location MV x is given by the