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Nonlinear behaviors of thin-walled Euler-Bernoulli beams with varying rotating speed which are attached to a rigid hub are investigated. Centrifugal force, aerodynamic load and the perturbed angular speed due to the inconstant air velocity are considered. The nonlinear factors are involved in displacement-strain relationships. The nonlinear governing partial differential equations of high-speed rotating thin-walled beam are established by using Hamiltonian Principle. Then, the ordinary differential equations of the rotating thin-walled beam are obtained by employing Galekin's approach during which Galekin's modes satisfy corresponding boundary conditions. The four-dimensional nonlinear averaged equations are obtained by applying the method of multiple scales. In this paper, the case of 1∶1 internal resonance is only considered. The results of the numerical simulation show that there exits complicated nonlinear behaviors in thin-walled Euler-Bernoulli beams with varying rotating speed.

In this paper a method is presented for computing the simplest normal form for a vector field associated with the singularity of two non-semisimple double zero eigenvalues. The method developed here uses the lower order nonlinear terms in the normal form for the simplifications of higher order terms. An explicit formulae are derived, which can be used to compute the coefficients of the simplest normal form and the associated nonlinear transformation. The theoretical model for the nonlinear oscillation of a simply supported rectangular thin plate is given to demonstrate the computational efficiency of the method.

In this paper, a new family of combined iterative methods for the solution of nonlinear equations is presented.The new family of methods is based on Newton's method and the family of sixth-order iterative methods developed by Chun. Per iteration the new methods require three evaluations of the function and two evaluations of its first derivative. Numerical tests show that it takes less number of iterations than Newton's method and some methods with third-order convergence. It is found that it only adds evaluation of the function at another point but its convergence order will be increased (p+1)-order above the original level.