17Approximate Analytical Methods


The exact solutions of problems associated with the free and forced vibration of continuous systems have been considered in earlier chapters. Exact solutions are usually represented by an infinite series expressed in terms of the normal or principal modes of vibration. In many practical applications, the solution of the vibration problem is dominated by the first few low‐frequency modes, and the effect of high‐frequency modes is negligible. In such cases the solution may be expressed in terms of a finite number of normal modes or in terms of assumed polynomials that describe the deformation shape of the continuous system. Exact solutions are possible only in relatively few simple cases of continuous systems. The exact solutions are particularly difficult to find for two‐ and three‐dimensional problems. Exact solutions are often desirable because they provide valuable insight into the behavior of the system through ready access to the natural frequencies and mode shapes.

Most of the continuous systems considered in earlier chapters have uniform mass and stiffness distributions and simple boundary conditions. However, some vibration problems may pose insurmountable difficulties either because the governing differential equation is difficult to solve or the boundary conditions may be extremely difficult or impossible to satisfy. In such cases we may be satisfied with an approximate solution of the vibration problem. Several methods ...

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