Enlarging the convergence region of the variable coefficient harmonic balance method for quasi-periodic solutions: applications to rotor systems

In rotating machinery, quasi-periodic vibrations can arise from the coexistence of synchronous and asynchronous components; therefore, accurate and computationally efficient prediction at the design stage is crucial. The Variable Coefficient Harmonic Balance Method (VCHBM) can efficiently compute quasi-periodic solutions. However, the solution of the resulting nonlinear algebraic equations strongly depends on the initial guess. In particular, estimating an unknown quasi-periodic frequency requires highly accurate results from direct numerical integration, which constitutes a practical bottleneck. In this study, to improve the initial-guess search in the VCHBM, we introduce the Levenberg–Marquardt method to update the Fourier coefficients. We further formulate the update of the unknown quasi-periodic frequency as a fixed-point iteration based on residual-norm minimization and accelerate its convergence using Anderson acceleration. Applications to an internally damped Jeffcott rotor and a turbocharger model supported by semi-floating ring bearings show that the proposed method converges stably from a wider range of initial estimates than the conventional Newton–Raphson method and accurately reproduces quasi-periodic steady-state solutions consistent with direct numerical integration (Runge–Kutta). In addition, we demonstrate that this method can trace solution branches, including unstable solutions and bifurcation points, which are difficult to capture via numerical integration.