1755 Conditions for the planes of symmetry of an elastically supported rigid body S J Jang 1 and Y J Choi 2∗ 1 Institute of Sound andVibration Research, University of Southampton, Southampton, Hampshire, UK 2 Department of Mechanical Engineering,Yonsei University, Seoul, Republic of Korea The manuscript was received on 22 October 2008 and was accepted after revision for publication on 5 February 2009. DOI: 10.1243/09544062JMES1406 Abstract: Introducing the planes of symmetry into an oscillating rigid body suspended by springs simplifies the complexity of the equations of motion and decouples the modes of vibration into in-plane and out-of-plane modes. There have been some research results from the investigation into the conditions for planes of symmetry in which prior conditions for the simplification of the equations of motion are required. In this article, the conditions for the planes of symmetry that do not need prior conditions for simplification are presented. The conditions are derived from direct expansions of eigenvalue problems for stiffness and mass matrices that are expressed in terms of in-plane and out-of-plane modes and the orthogonality condition with respect to the mass matrix. Two special points, the planar couple point and the perpendicular translation point are identified, where the expressions for stiffness and compliance matrices can be greatly simplified. The simplified expressions are utilized to obtain the analytical expressions for the axes of vibration of a vibration system with planes of symmetry. Keywords: planes of symmetry, screw theory, analytic solution 1 INTRODUCTION An elastically supported rigid body has the com- plex equations of motion in general where trans- lational and rotational displacements are coupled together. The equations of motion can be simplified and grouped when some decoupling conditions are applied. Derby [1] proposed the condition for decou- pling translations from rotations when a rigid body has four isolators. Himelblau and Rubin [2] simplified the equations of motion of an oscillating body by apply- ing some prior conditions and derived the condition for planes of symmetry. Racca [3] used the focused isolators to make the six modes of vibration decou- pled from each other. Xuefeng [4] presented a detailed mathematical description of a decoupling arrange- ment of engine mounts. Jeong and Singh [5] compared ∗ Corresponding author: Department of Mechanical Engineering, Yonsei University, 262 Seongsanno, Seodaemun-gu, Seoul 120-749, Republic of Korea. email: yjchoi@yonsei.ac.kr; ychoi1260@gmail.com the existing decoupling schemes with the torque roll axis method. In this article, screw theory is utilized to express the vibration of an elastically supported rigid body. The vibration modes of an elastically suspended rigid body were first viewed as screws by Ball [6]. Dimentberg [7] expressed the equations of motion for an oscillat- ing body in terms of screws. Blanchet and Lipkin [8] investigated a linear vibration of a planar sys- tem using screw theory and derived the cubic equa- tions for three vibration modes that they termed the vibration centres. The plane of symmetry of a vibrating system is a plane about which the vibration modes are decoupled into in-plane and out-of-plane modes. The in-plane mode refers to the vibration of a body about an axis perpendicular to the plane of symmetry, while the out- of-plane mode means the vibration about an axis lying on the plane. These interpretations can provide some- what transparent geometrical understanding about vibration modes. Dan and Choi [9, 10] showed that the planes of symmetry depend on the location of the centre of elasticity via screw theory. They derived the analytical solutions for the axes of vibrations and JMES1406 © IMechE 2009 Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science