IN recent years much attention has been devoted to devising methods for the numerical solution of Lagrangian frequency equations in practical problems. A powerful method based on iteration was given by Duncan and Collar in the Phil. Mag. for May, 1934. This method, which was described by one of the present authors in AIRCRAFT ENGINEERING, Vol. XIV, April, 1942, pp. 108–110, is essentially one of successive approximation and lias the advantage of simultancously giving the modes associated with the frequencies. First the highest root is found together with its associated modes. Then by an ingenious artifice the equations are reduced by one degree being freed from the root found ; and so on. A defect of the method is due to the fact that the accuracy diminishes with each successive root. Consequently it becomes necessary to start the computation with a large number of significant figures in order to achieve the requisite accuracy as the number of roots to be found increases. There is another drawback to the method in that it cannot be applied to the general Lagrangian equations direct. They have first to be recast into “ canonical ” form in which the root to be found appears only along the “ leading diagonal”. This involves solutions of simultaneous equations with the same number of variables as the number of roots in the original equations.
Morris, J. and Head, J.W. (1942), "Lagrangian Frequency Equations: An “Escalator” Method for Numerical Solution", Aircraft Engineering and Aerospace Technology, Vol. 14 No. 11, pp. 310-316. https://doi.org/10.1108/eb030958
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