Computation of Normal Forms for Eight-Dimensional Nonlinear Dynamical System and Application to a Viscoelastic Moving Belt

摘要

An improved adjoint operator method is employed to compute third order normal forms of an eight-dimensional nonlinear dynamical system and the associated nonlinear transformation for the first time. Two different cases which respectively are four pairs of pure imaginary eigenvalues and one non-semisimple double zero and three pairs of pure imaginary eigenvalues for the linear operator are considered in the eight-dimensional nonlinear dynamical system. First, an improved adjoint operator method is described in detail by analyzing the eight-dimensional nonlinear dynamical system. The formulae of computing third order normal forms of the eight-dimensional nonlinear system are derived and the Maple symbolic program is developed in two different cases of the linear operator. Then, third order normal forms and their coefficients for the eight-dimensional nonlinear dynamical system in two different cases of the linear operator are obtained by executing the Maple program. The relationship between the coefficients of normal forms and ones of the original nonlinear systems is given. Finally, this approach is also applied to obtain normal form of the averaged equation for a parametrically excited viscoelastic moving belt. The results obtained here indicate that this improved adjoint operator method is a convenient and efficient approach to obtain normal forms of higher dimensional nonlinear dynamical systems. It is also shown that we may respectively obtain normal forms, their coefficients and the associated near identity nonlinear transformations for the eight-dimensional nonlinear system in two different cases by using a same main Maple symbolic program.