Abstract
Nonlinear vibration problems are reviewed in their aspect of their phenomenon and their basic nonlinear equations of motion. Many papers which deal with nonlinear vibrations in elastic systems depend on the Duffing equation or the Mathieu-Hill equation. But they do not include the general nonlinear equations of motion obtained by considering the finite deformation theory in elasticity, because the Duffing equation corresponds to the system which nonlinear springs are monotone functions of displacement and the Mathieu-Hill equation is the approximate equation derived by assuming separate variables. Then if we adopt the general nonlinear equations of motion as the basic equation for nonlinear vibrations, the treatment of nonlinear vibration problems is unified. In order to classify nonlinear vibrations it is necessary to grasp the global feature of unkowns in algebraic equations. The local feature of them have been precisely studied in the static nonlinear stability problems. Then applying the theory we construct the classification of the unkowns in algebraic equation and a global definition of the symmetric bifurcation.
