The concept of nonlinear modes applied to friction-damped systems

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Are Chebyshev-based stability analysis and Urabe's error bound useful features for Harmonic Balance?

Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors' knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe's error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.Comment: The final version of this article is available online at https://doi.org/10.1016/j.ymssp.2023.11026

Fully Coupled Forced Response Analysis of Nonlinear Turbine Blade Vibrations in the Frequency Domain

For the first time, a fully-coupled Harmonic Balance method is developed for the forced response of turbomachinery blades. The method is applied to a state-of-the-art model of a turbine bladed disk with interlocked shrouds subjected to wake-induced loading. The recurrent opening and closing of the pre-loaded shroud contact causes a softening effect, leading to turning points in the amplitude-frequency curve near resonance. Therefore, the coupled solver is embedded into a numerical path continuation framework. Two variants are developed: the coupled continuation of the solution path, and the coupled re-iteration of selected solution points. While the re-iteration variant is slightly more costly per solution point, it has the important advantage that it can be run completely in parallel, which substantially reduces the wall clock time. It is shown that wake- and vibration-induced flow fields do not linearly superimpose, leading to a severe underestimation of the resonant vibration level by the influence-coefficient-based state-of-the-art methods (which rely on this linearity assumption).Comment: 24 pages, 14 figures, preprint submitted to Journal of Computers and Structure