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This paper proposes a geometric approach to the conditions for mode decoupling of a vibration system of an elastically supported single rigid body and presents the conditions that the system has only pure rotation modes of vibration. A small oscillation of a rigid body is indeed a repetitive screw motion and thus vibration modes are expressed by screws in general, which results in the difficulty involved in solving a vibration problem. The complexity of a vibration system can be alleviated for both analysis and synthesis if the system has only rotation modes. In order to acquire the decoupling techniques, this paper begins by investigating a stiffness matrix which can be separated into the sum of two rank 3 stiffness matrices, which are realizable by using co-reciprocal line vectors. From the co-reciprocity, the separable stiffness matrix can be regarded as a linear transformation between two 3-systems of screws containing only line vectors. Using the properties of the linear transformation and the screw systems, the conditions for mode decoupling, or the conditions for only pure rotation modes are derived and described by geometric relations between inertia and stiffness, and three cases of vibration systems with simple geometric nature are identified.

The Conditions for a Linear Vibration System to Have Only Pure Rotation Modes