Galerkin-Ivanov变换用于连续结构碰撞振动的非光滑模拟
Galerkin-Ivanov transformation for nonsmooth modeling of vibro-impacts in continuous structures
摘要：
本文研究了连续结构与刚性分布障碍物碰撞的非光滑振动冲击运动。Galerkin方法被用来近似的结构，这导致在一个系统的常微分方程（ODE）的偏微分方程的解决方案。当这些常微分方程受到单边约束和速度跳跃条件时，必须使用事件检测算法来精确计算碰撞时间。在存在多个同时撞击的情况下的事件检测是计算要求高的任务。Ivanov提出了一种受单边约束的多自由度碰撞振动系统的非光滑变换。这种转换消除了问题的单边约束，因此，在数值积分过程中不需要事件检测。伊万诺夫利用他的变换对振动碰撞运动的稳定性和分叉进行了分析计算;然而，他没有探索其在空间连续结构中模拟分布式碰撞的应用。我们采用Ivanov变换来处理空间连续结构中的多个单边约束。此外，在模态坐标中精确地施加速度跳跃条件是不平凡的和具有挑战性的。因此，在这项工作中，我们使用一个模态物理转换系统从模态到物理坐标的空间离散网格。然后，我们将Ivanov变换应用于物理系统来模拟结构的振动-碰撞运动。所开发的方法证明了通过建模对刚性分布表面的非线性字符串的分布式碰撞。为了验证，我们将我们的结果与众所周知的惩罚方法进行比较。
英文摘要：
This work deals with the modeling of nonsmooth vibro-impact motion of a continuous structure against a rigid distributed obstacle. Galerkin's approach is used to approximate the solutions of the governing partial differential equations of the structure, which results in a system of ordinary differential equations (ODEs). When these ODEs are subjected to unilateral constraints and velocity jump conditions, one must use an event detection algorithm to calculate the time of impact accurately. Event detection in the presence of multiple simultaneous impacts is a computationally demanding task. Ivanov proposed a nonsmooth transformation for a vibro-impacting multi-degree-of-freedom system subjected to a single unilateral constraint. This transformation eliminates the unilateral constraints from the problem and, therefore, no event detection is required during numerical integration. Ivanov used his transformation to make analytical calculations for the stability and bifurcations of vibro-impacting motions; however, he did not explore its application for simulating distributed collisions in spatially continuous structures. We adopt Ivanov's transformation to deal with multiple unilateral constraints in spatially continuous structures. Also, imposing the velocity jump conditions exactly in the modal coordinates is nontrivial and challenging. Therefore, in this work we use a modal-physical transformation to convert the system from modal to physical coordinates on a spatially discretized grid. We then apply Ivanov's transformation on the physical system to simulate the vibro-impact motion of the structure. The developed method is demonstrated by modeling the distributed collision of a nonlinear string against a rigid distributed surface. For validation, we compare our results with the well-known penalty approach.
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