Keywords: composite bar, coupled motion, natural frequencies, exact coupling relationships, transcendental eigenvalue problem, Wittrick-Williams algorithm.

This study uses an equivalent continuum approach to develop two forms of exact solution for calculating the materially coupled natural frequencies of a family of simple bar elements whose motions exhibit either extension-shear, shear-twist or extension-twist coupling. Initially, the differential equations governing the extension-shear, coupled motion of an equivalent continuum member are developed from first principles. A common solution procedure then leads to either an exact dynamic stiffness matrix or an exact relational model that links the uncoupled natural frequencies to the coupled ones that stem from them.

Such an exact approach, which allows for the uniform distribution of mass in the member, necessitates the solution of a transcendental eigenvalue problem. This is achieved using the Wittrick-Williams algorithm, which guarantees that the required natural frequencies are converged upon to any desired accuracy with the certain knowledge that none have been missed.

Under normal circumstances, the coupling coefficients that create the coupled motion will be equal and this leads to both a symmetric stiffness matrix and the normal form of the relational model. Theoretically, however, there still remains the possibility that the coupling coefficients are not equal. If this were the case, the theory would violate the self adjoint requirement, and hence Maxwell's reciprocal theorem for linear structures. However, this option has been retained for two reasons. Firstly, because a lack of reciprocity is apparent in the motion of a number of simple components, such as helical springs, wire ropes etc.; and secondly because the relational model still leads to an elegant and simple exact solution that may well find useful application elsewhere.

In overview, both the stiffness and relational formulations can be used in similar ways to build up and analyse more complex members. In general, the stiffness approach is somewhat more versatile, but becomes largely impractical when the coupling coefficients yield a stiffness matrix that is not self adjoint. In this case the transcendental nature of the matrix is difficult to deal with, since the Wittrick-Williams algorithm cannot be applied in such circumstances. The relational model, on the other hand, deals with this case simply and efficiently by first calculating the uncoupled frequencies of the bar, or system of bars, which can be achieved easily without recourse to the coupling coefficients. Only then are the coupling coefficients imposed to yield the required solutions. As a result, the Wittrick-Williams algorithm can be used freely to establish the initial uncoupled frequencies. An additional advantage of the relational model is that it offers an easy way of establishing the actual value of the coupling coefficients from experimental results.

A number of examples are given to confirm the accuracy of the approach and indicate its range of application.

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