Liu X, Wagg D J
Nuclear Advanced Manufacturing Research Centre, University of Sheffield, Rotherham S60 5WG, UK.
Department of Mechanical Engineering, University of Sheffield, Sheffield S1 3JD, UK.
Proc Math Phys Eng Sci. 2019 Aug;475(2228):20190042. doi: 10.1098/rspa.2019.0042. Epub 2019 Aug 7.
In this paper, we describe a direct normal form decomposition for systems of coupled nonlinear oscillators. We demonstrate how the order of the system can be reduced during this type of normal form transformation process. Two specific examples are considered to demonstrate particular challenges that can occur in this type of analysis. The first is a 2 d.f. system with both quadratic and cubic nonlinearities, where there is no internal resonance, but the nonlinear terms are not necessarily -order small. To obtain an accurate solution, the direct normal form expansion is extended to -order to capture the nonlinear dynamic behaviour, while simultaneously reducing the order of the system from 2 to 1 d.f. The second example is a thin plate with nonlinearities that are -order small, but with an internal resonance in the set of ordinary differential equations used to model the low-frequency vibration response of the system. In this case, we show how a direct normal form transformation can be applied to further reduce the order of the system while simultaneously obtaining the normal form, which is used as a model for the internal resonance. The results are verified by comparison with numerically computed results using a continuation software.
在本文中，我们描述了耦合非线性振子系统的一种直接范式分解。我们展示了在这种范式变换过程中系统的阶数如何能够降低。考虑了两个具体例子以说明在这类分析中可能出现的特殊挑战。第一个是具有二次和三次非线性的二自由度系统，其中不存在内共振，但非线性项不一定是高阶小量。为了获得精确解，将直接范式展开扩展到高阶以捕捉非线性动态行为，同时将系统的阶数从二自由度降低到一自由度。第二个例子是一个薄板，其非线性是高阶小量，但在用于对系统低频振动响应进行建模的常微分方程组中有内共振。在这种情况下，我们展示了如何应用直接范式变换来进一步降低系统的阶数，同时获得范式，该范式用作内共振的模型。通过与使用延拓软件的数值计算结果进行比较来验证结果。
