Augmenting the Harmonic Balance Method by Stability Analysis and Error Estimation and its Application to Vibro-Impact Processes

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The high-order Harmonic Balance (HB) method reduces the effort for simulating nonlinear vibrations often by 2-4 orders of magnitude compared to numerical integration. Only with this substantial reduction, the thorough analysis needed to design many nonlinear vibrating systems becomes feasible. However, HB has two major limitations: unreliable stability analysis and lacking error estimation. Therefore, the first objective is to make HB a reliable method by augmenting it accordingly. To permit statements on the asymptotic stability, the monodromy matrix is determined, however, the currently required numerical integration will be replaced by the computationally much more efficient solution of a linear algebraic equation system using the properties of Chebyshev polynomials and exploiting insight into the underlying mechanical problem. The theoretical basis for the error estimation will be Urabe's theorem from 1965. This way, the engineering value of his mathematical result is analyzed for the first time. The developed computational capabilities are validated and compared to the state of the art.The second objective is to gain deeper insight into the nonlinear dynamic behavior of Vibro-Impact Systems with closely spaced modes. Such systems are challenging for any computational method, as impacts can trigger a strong energy exchange among the vibration modes, and the severe nonlinearity gives rise to intricate types of stability loss. These are ideal conditions for analyzing not only the opportunities but also the limitations of the augmented HB method. Preliminary numerical analyses indicate that such systems have isolated regimes (isola) of stable high-level responses. An important goal will be to understand under what conditions such isola occur. This will be analyzed using advanced numerical and experimental methods. This way, for the first time, empirical evidence of isola is provided for a system with closely spaced modes.The stability analysis and error estimation are crucial for filtering out the physically relevant responses from the numerical solutions, and to predict the emergence of (otherwise undetected) new vibration regimes. The error estimation is also crucial for constructing mathematically rigorous techniques of harmonic-order-refinement. Only with the proposed methodological developments, HB becomes a reliable method for nonlinear vibrations. This way, the proposed project makes a strong contribution to facilitating the paradigm shift from the avoidance to the intentional use of nonlinearity in the design of vibrating systems.