This paper presents and proves an algorithm for determining the natural undamped frequencies of vibration of any linearly elastic structure if its dynamic stiffness matrix K(ω0) corresponding to any finite set of displacements D is known. In general K(ω) is not a linear function of ω2, and methods which are available for solving linear eigenvalue problems are inapplicable. The algorithm is valid for systems with either a finite or infinite number of degrees of freedom. It enables one to calculate how many natural frequencies lie below any chosen frequency, without determining them, and hence to converge on any required natural frequency to any specified accuracy. Coincident natural frequencies, and exceptional ones which correspond to D = 0 and not to det K (ω) = 0, are automatically accounted for. The algorithm is likely to be applied mainly to systems with an infinite number of degrees of freedom, for which there is no comparable approach, and numerical results to illustrate such an application are presented. It is possible however, that there may be cases of large finite systems for which it would be advantageous to use the algorithm, rather than Householder's method for example, since it enables the structure to be broken down into sub-structures and full advantage to be taken of any that are indentical.
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Wittrick et al. (1971) studied this question.