Numerical solution of the dynamic stability problems

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Abstract
A novel integral equation technique is employed for the analysis of dynamic stability problems. The governing equation of the linearized parametric resonance problem is transformed into an integral equation. The kernel of the integral equation is computed as the influence function for the deflection and/or bending moment of a corresponding beam.The highest derivative of the governing function (in our case fourth derivative of the displacement function) is chosen as the basic unknown. Using the formal analogy with the differential equation of the beam flexure this highest derivative is comprehended as some unknown transverse ‘load’. The distribution of this ‘load’ is a priori assumed to be polygonal. Using elementary methods of structural analysis, the displacements due to the assumed ‘load’ are determined. These displacements, arrayed into a square matrix, approximate the kernel of the governing integral equation.The subsequent procedure via Hill's determinant is a conventional one. The results prove to be accurate enough even for a very modest number of points of integration. This reflects the fact that the method is based on numerical integration rather than on numerical differentiation.

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