Many modern engineering structures exhibit nonlinear vibration. Characterizing such vibrations efficiently is critical to optimizing designs for reliability and performance. For linear systems, steady-state vibration occurs only at the forcing frequencies. However, nonlinearities (e.g., contact, friction, large deformation, etc.) can result in nonlinear vibration behavior including superharmonics — responses at integer multiples of the forcing frequency. When the forcing frequency is near an integer fraction of the natural frequency, superharmonic resonance occurs, and the magnitude of the superharmonics can exceed that of the fundamental harmonic that is externally forced. Characterizing such superharmonic resonances is critical to improving engineering designs. The present work extends the concept of phase resonance nonlinear modes (PRNM) to be applicable to general nonlinearities, and is demonstrated for eight different nonlinear forces. The considered forces include stiffening, softening, contact, damping, and frictional nonlinearities that have not been previously considered with PRNM. The proposed variable phase resonance nonlinear modes (VPRNM) method can accurately track superharmonic resonances for hysteretic nonlinearities that exhibit amplitude dependent phase resonance conditions that cannot be captured by PRNM. The proposed method allows for characterization of superharmonic resonances without constructing a full frequency response curve at every force level with the harmonic balance method. Thus, the present method allows for analysis of potential failures due to large amplitudes near the superharmonic resonance with reduced computational cost. The consideration of single degree of freedom systems in the present paper provides insights into superharmonic resonances and a basis for understanding internal resonances for multiple degree of freedom systems.

中文翻译：

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跟踪保守和迟滞单自由度系统非线性振动的超谐波共振

许多现代工程结构表现出非线性振动。有效表征此类振动对于优化设计的可靠性和性能至关重要。对于线性系统，稳态振动仅发生在强制频率下。然而，非线性（例如，接触、摩擦、大变形等）可能导致非线性振动行为，包括超谐波——在强制频率的整数倍处的响应。当强制频率接近固有频率的整数分之一时，会发生超谐波共振，并且超谐波的幅度可以超过外加的基波谐波的幅度。表征这种超谐波共振对于改进工程设计至关重要。目前的工作扩展了相位谐振非线性模式（PRNM）的概念，使其适用于一般非线性，并针对八种不同的非线性力进行了演示。考虑的力包括以前 PRNM 未考虑过的刚度、软化、接触、阻尼和摩擦非线性。所提出的可变相位谐振非线性模式 (VPRNM) 方法可以准确跟踪滞后非线性的超谐波谐振，这些滞后非线性表现出 PRNM 无法捕获的振幅相关相位谐振条件。所提出的方法允许表征超谐波谐振，而无需使用谐波平衡方法在每个力水平构建完整的频率响应曲线。因此，本方法允许以降低的计算成本来分析由于超谐波谐振附近的大振幅导致的潜在故障。本文对单自由度系统的考虑提供了对超谐波共振的见解，并为理解多自由度系统的内部共振奠定了基础。