Orthogonality of modes of structures when using the exact transcendental stiffness matrix method

This paper presents theory, physical insight and results for mode orthogona lity of piecewise continuous structures, including both coincident and non- coincident natural frequencies. The structures are ones for which exact mem ber equations have been obtained by solving the governing differential equa tions, e.g. as can be done for members of plane frames or prismatic plate a ssemblies. Such member equations are transcendental functions of the distri buted member mass and the frequency. They are used to obtain a transcendent al overall stiffness matrix for the structure, from which the natural frequ encies are extracted by using the Wittrick-Williams algorithm, prior to usi ng any existing method to find the modes which are examined from the orthog onality viewpoint in this paper. The natural frequencies and modes found ar e the exact values for the structure in the sense that the usual finite ele ment method approximations are avoided.