Keywords: exact dynamic stiffness matrix, Wittrick-Williams algorithm, elastically supported structures.

The dynamics of a family of simple, but extremely useful structural elements is governed by a linear second order differential equation. This equation allows for the uniform distribution of mass and stiffness and enables the motion of strings and shear beams, together with the axial and torsional motion of bars to be described exactly. As a result, each member type in this family has been treated exhaustively when considered as a single member or when joined contiguously to others. However, when such members are linked in parallel by uniformly distributed elastic interfaces, their complexity becomes significantly more intractable and it is this class of structures that has led to renewed interest. Consideration is therefore given herein to determining the exact natural frequencies of such structures, where each member can have different material and geometric properties and be supported independently on distributed elastic foundations. The formulation is general and applies to any member whose motion is governed by such an equation.

Initially, differential equations governing the coupled motion of the system are developed from first principles. A common solution procedure then leads either to an exact dynamic stiffness matrix or to a series of exact relational models that link the uncoupled frequencies to the coupled ones that stem from them. The former theory is enhanced by providing an appropriate formulation of the Wittrick-Williams algorithm, which guarantees that any required frequency can be calculated to any desired accuracy with the certain knowledge that none have been missed. When appropriate symmetry is present, the theory can be extended to cover three and four member structures.

The approach adopted differs from all previous work in three distinctly different ways. Firstly, it is based on an exact dynamic stiffness approach. This is important because, at the heart of any exact solution procedure for this type of structure, it is necessary to solve a transcendental eigenvalue problem. Currently this can only be achieved exactly by using an exact dynamic stiffness formulation in conjunction with the Wittrick-Williams algorithm. Secondly, a comprehensive set of inter-relationships between the natural frequencies of the component elements comprising the structure have been formulated using a simple and novel procedure. Finally, the theory has been developed in the context of two linked shear beams. This has not been done previously and leads directly to the possibility of using well established simplification procedures that reduce multi-bay, multi-storey sway frames to equivalent one bay frames and then to even simpler global models that can often retain sufficient accuracy for preliminary analysis and design procedures.

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