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

高阶谐波平衡（HB）方法与数值积分相比，通常可将模拟非线性振动的工作量减少2-4个数量级。只有这样，设计许多非线性振动系统所需的全面分析才变得可行。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使其成为可靠的方法。为了允许声明的渐近稳定性，单值矩阵的确定，然而，目前所需的数值积分将被取代的计算更有效的解决方案的线性代数方程系统使用的性质切比雪夫多项式和利用洞察到潜在的机械问题。误差估计的理论基础是1965年的浦部定理。首次分析了其数学成果的工程价值。开发的计算能力进行验证，并与最先进的。第二个目标是更深入地了解振动冲击系统的非线性动力学行为与紧密间隔的模式。这种系统对于任何计算方法都是具有挑战性的，因为冲击可以触发振动模式之间的强烈能量交换，并且严重的非线性引起复杂类型的稳定性损失。这些都是理想的条件，不仅分析的机会，但也扩大HB方法的局限性。初步的数值分析表明，这样的系统有孤立的制度（isola）的稳定的高层次的反应。一个重要的目标将是了解在什么条件下发生这种孤立。这将使用先进的数值和实验方法进行分析。这样，第一次，孤立的经验证据提供了一个系统的紧密间隔modes.The稳定性分析和误差估计是至关重要的过滤出的物理相关的响应从数值解，并预测出现（否则未检测到）新的振动制度。误差估计对于构造数学上严格的谐波阶次精化技术也是至关重要的。只有随着提出的方法的发展，HB才成为非线性振动的可靠方法。通过这种方式，拟议的项目作出了强有力的贡献，以促进从避免非线性振动系统的设计有意使用的范式转变。

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.