Free and steady state vibration of non‐linear structures using a finite element–non‐linear eigenvalue technique

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AbstractThe problem of free vibration of non‐linear structures is considered initially. It is shown that this problem can be represented as a non‐linear eigenvalue problem. Variational principles for non‐linear eigenvalue problems are defined. These variational principles are implemented with finite element models to define numerical approximations for the free vibration problem. The solution of these approximate equations provides a set of non‐linear modal vectors and natural frequencies which vary with the amplitude of the solution. The non‐linear eigenvalue parameters can be used in modal expansion approximations for the non‐linear transient or steady state response of structural systems. To demonstrate the proposed techniques the free vibration and steady state vibration characteristics of a geometrically non‐linear circular plate are determined.