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

与数值积分相比，高次谐波平衡法(HB)减少了模拟非线性振动的工作量2-4个数量级。只有这样大幅度减少，设计许多非线性振动系统所需的透彻分析才变得可行。然而，HB有两个主要的局限性：不可靠的稳定性分析和缺乏误差估计。因此，第一个目标是通过相应地增强HB来使其成为一种可靠的方法。为了允许关于渐近稳定性的声明，确定了单调矩阵，然而，当前所需的数值积分将被利用切比雪夫多项式的性质并深入了解基本力学问题的线性代数方程组的计算效率更高的解所取代。误差估计的理论基础将是1965年以来的Urabe定理。这样，第一次分析了他的数学结果的工程价值。对开发的计算能力进行了验证，并与ART的状态进行了比较。第二个目标是更深入地了解近距离振动冲击系统的非线性动力学行为。这类系统对任何计算方法都具有挑战性，因为冲击会引发振动模式之间的强烈能量交换，并且严重的非线性会导致复杂类型的稳定性损失。这些都是分析增广HB方法的机会和局限性的理想条件。初步的数值分析表明，此类系统具有稳定的高层响应的孤立区域(ISLA)。一个重要的目标将是了解在什么情况下会发生这种Isola。这将使用先进的数值和实验方法进行分析。稳定性分析和误差估计对于从数值解中滤除物理上相关的响应和预测新振型的出现(否则未被检测到)至关重要。误差估计对于构造数学上严格的调和阶次细化技术也是至关重要的。只有随着所提出的方法的发展，HB才成为研究非线性振动的可靠方法。通过这种方式，建议的项目对促进振动系统设计中从避免使用非线性到有意使用非线性的范式转换做出了巨大贡献。

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.