SUMMARYA multiharmonic method for analysis of non-linear dynamic systems submitted to periodic loading conditions is presented. The approach is based on a systematic use of the fast Fourier transform. The exact linearization of the resulting equations in the frequency domain allows to obtain full quadratic convergence rate.

We present an implementation of the multiharmonic balance method (MHB) where intensive use of the Fast Fourier Transform algorithm (FFT) is made at all stages of calculations. The MHB method is not modi®ed in essence, but computations are organized to obtain a very attractive method that can be applied systematically on general nonlinear vibration problems. The resulting nonlinear algebraic problem is solved by a particular implementation of a continuation method. Nonlinear vibration results are analyzed a posteriori by a Floquet method to determine their stability. The technique is applied on a series of problems of different nature, demonstrating the robustness and¯exibility of the approach. IntroductionThe multi-harmonic balance method (MHB) has been widely used to solve nonlinear vibration problems under periodic excitation. It ®nds applications in several ®elds of mechanical engineering, e.g. machine dynamics, vehicle dynamics, helicopter rotor blade analysis, structural dynamics.The method is well-known from literature. Urabe [1] investigated the convergence conditions of the method and presented numerical applications [2] using the classical Fourier transform. Lau et al. [3±5] and more recently [6] developed an incremental form of the method. Pierre et al. [7, 8] and Ferri [9] followed the same approach to solve nonlinear vibration problems involving dry-friction. Ling and Wu [10] introduced the use of the Fast Fourier Transform (FFT) algorithm in Urabe's formulation. The same path was followed by other authors [11±16]. The use of FFT permits an overwhelming gain of CPU time when computing the Fourier transform. Most authors use the FFT to switch displacements, velocities, accelerations and forces between the time and frequency domains. However, even if the original proposal by Urabe included the analytical expression of the Jacobian of the nonlinear algebraic problem, most of them avoid computing this matrix. Ling and Wu [10] used a Broyden method; Cameron and Grif®n [11] used either a Picard iteration or a Jacobian matrix evaluated by ®nite differences; Lewandowski [14] gave particular expressions for the contributions of nonlinear terms to the Jacobian for the case of geometric nonlinearities in the form of trigonometric expansions.In reference [15] we presented an approach which differs from most others in that an analytical expression for the Jacobian matrix of the nonlinear algebraic problem is developed for general applications. This fact allows to solve the nonlinear algebraic problem with utmost ef®-ciency, reaching quadratic convergence rate. We have shown how the Jacobian matrix can be computed from the Fourier transform of the time domain stiffness, damping and mass matrices of the system under analysis.The MHB leads to a nonlinear algebraic problem in which the solution should be searched in the Fourier transformed displacements versus excitation period space. The solution is found in the form of a nonlinear dynamic equilibrium path in this space for varying values of the excitation period. St...