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

This paper presents theory, physical insight and results for mode orthogonality of piecewise continuous structures, including both coincident and non-coincident natural frequencies. The structures are ones for which exact member equations have been obtained by solving the governing differential equations, e.g. as can be done for members of plane frames or prismatic plate assemblies. Such member equations are transcendental functions of the distributed member mass and the frequency. They are used to obtain a transcendental overall stiffness matrix for the structure, from which the natural frequencies are extracted by using the Wittrick-Williams algorithm, prior to using any existing method to find the modes which are examined from the orthogonality viewpoint in this paper. The natural frequencies and modes found are the exact values for the structure in the sense that the usual finite element method approximations are avoided.