A Matrix Solution for the Vibration Modes of Nonuniform Disks

Abstract

Abstract An arbitrary disk is represented by a simulated disk composed of circumferential strips. Alternate strips are considered to be massless, constant-thickness elements with the average local elastic properties of the actual disk. Intermediate strips are considered to have the properties of local mass and polar moment of inertia, but to have no physical dimensions or elasticity. A matrix vector, formed of the local antinodal value of deflection, slope, moment, and transverse force, may be operated on by matrices representative of the elastic strips and by matrices representative of the vibratory inertia loading, centrifugal inertia loading, internal stress, and external supports at the mass strips. Thus the influence of boundary conditions at the outer edge on conditions at the inner edge may be calculated in a simple efficient manner. Successive guesses of vibration frequency lead to final satisfaction of all boundary conditions. Concise treatment of all types of boundary conditions and numerical values of required matrices are given in tables. The results of a sample calculation are compared with exact analytic results.