A harmonic balance solution for the intrinsic 1D nonlinear equations of the beams

Abstract

In this paper, a harmonic balance method is introduced to solve the intrinsic 1D nonlinear equations presented by Hodges for analyzing initially curved and twisted anisotropic rotating beams. First, the nonlinear first-order partial differential equations are discretized in space domain using the Galerkin approach. Then, the time response of the system of equations is approximated by using first and the second harmonic terms under the harmonic excitations applied to the structure. The harmonic balance algorithm has been developed for the linearized system of equations about the steady-state solution and for fully nonlinear equations. The differences between these two systems have been addressed under consideration of different levels of external loads. Fully analytical formulae have been presented for the HB solution of the linearized equations in this work. In the proposed approach for the nonlinear system, the Jacobian matrix utilized in the numerical solution part is obtained analytically to increase the time efficiency of the solution method. The results of the proposed approach for an initially twisted and curved blade are compared with a different algorithm based on a finite difference approach to discretize the equations and a time marching method to solve the system of equations in time domain. The comparison between the outputs of the harmonic balance method and the time marching method demonstrate high correlations between these distinct methods. It should be noted that the accuracy of the time marching method solution has been evaluated in another publication by the authors using the output of FLIGHTLAB for a blade with different types of boundary conditions. The proposed harmonic balance-based method delivers a very time-efficient approach for time analysis of twisted/curved beams and blades under harmonic loads in comparison with other employed methods. The developed approach can be a powerful computational tool for design optimization of composite blades.