Computation of normal forms for high dimensional non-linear systems and application to non-planar non-linear oscillations of a cantilever beam

A new and efficient computation of the normal forms is developed in this paper for high dimensional non-linear systems, and the computational method is applied to non-planar non-linear oscillations of a cantilever beam. The method developed here has the advantage of directly calculating the coefficients of the normal forms and the associated near identity non-linear transformations for three different cases, that is, (1) the case of two pairs of pure imaginary eigenvalues; (2) the case of one non-semisimple double zero and a pair of pure imaginary eigenvalues; and (3) the case of two non-semisimple double zero eigenvalues. The final partial differential equations of various resonant cases appear in a canonical form whose solutions can be conveniently obtained using polynomial equations. With the aid of the Maple software, a symbolic program for computing the normal forms of high dimensional non-linear systems is given. Comparing the method developed here with other methods of computing the normal forms, it is understood that we may, respectively, obtain the normal forms, the coefficients of the normal forms and the associated near identity non-linear transformations for three resonant cases by using a same main Maple symbolic program. Moreover, the method is easy to apply to engineering problems. The normal forms of the averaged equations and their coefficients for non-planar non-linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated for two resonant cases.