The task of establishing analytically the natural frequencies of vibration of a partially embedded beam gives rise to a particular difficulty because, frequently, the modes are clustered in close proximity to each other. Due to this, and the fact that the exact formulated solution is in the form of a non-symmetrical eigenvalue problem, neither of the established techniques (using either crude graphical methods or the infallible algorithmic methods which are based on a stiffness matrix approach) can be utilised directly in the solution of this problem. Consequently, an alternative technique is proposed which makes use of the concept of sign counting of the main diagonal of the upper triangular form of the solution matrix introduced in the earlier methods. An efficient algorithm is developed which allows near-infallible detection of eigenvalues by using a dynamic increment in the search for diagonal element sign pattern changes. This paper outlines the main features of this procedure and provides a typical example to illustrate its operation.

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