Coupled Vibrations of an Elastically Mounted Rigid Body with One Plane of Symmetry

Abstract

An elastically mounted in-line aircraft engine is treated as a rigid body with a vertical plane of symmetry. The symmetry permits the six equations of motion to be resolved into two independent groups of three each. By using dimensionally homogeneous variables and the methods of vector-matrix analysis, the three scalar equations in each group are reduced to one. All the inertial and stiffness properties of the engine and its mounting system are contained in a single matrix coefficient called the frequency matrix. The static displacement, natural frequencies, and the dynamic magnification factors are expressed as operations on the frequency matrix. A geometric interpretation is given for the modes of motion and the relative amplitude vectors. INTRODUCTION A i N IN-LINE AIRCRAFT ENGINE can be assumed to have a vertical plane of symmetry. I t is customary to mount the engine at four points lying in a plane normal to the plane of symmetry and containing the crankshaft centerline. In general, the center of mass of the engine will not lie in the plane of the mounting points, nor will the longitudinal principal central axis be parallel to this plane. Thus, there will be coupling terms in all six of the equations of motion. By a suitable choice of the elastic properties of the mounting devices it is possible to make some of the coupling terms vanish. I t is not practicable to make them all vanish. It is desirable to have a general solution describing the modes of motion and giving the natural .frequencies, the displacements, and the dynamic magnification factors. This problem has been discussed by von Schlippe. The solution presented here is based on a generalization of the simple linear vibration equation x + con x 0 (a) and is suitable for routine calculation. Only the case of a propeller at rest will be considered here. Maximum simplicity ot the equations of motion is achieved by using the principal central axes for reference. The positive directions of the three linear and three angular coordinates are shown in Fig. 1. Because of the plane of symmetry (XZ) the equations of motion consist of two independent sets of three each. The equations containing the coordinates x, 6, z describe the symmetric modes of motion, and the equaReceived September 6, 1946. * Based on research proposed by the Aircraft Laboratory, Wright Field, and carried out by the author under a contract with the A.A.F. Air Materiel Command. t Associate Professor of Mechanics. tions containing the coordinates , y, \p describe the antisymmetric modes of motion.