Resonance phenomena in impacting systems can be defined as an amplitude increasing during periodically applied impacts. The wave cancellation phenomenon is defined as application of certain conditions to cancel the wave fully. The double impact system is defined as the application of the first impact with a certain duration $$\tau $$ τ and then the application of a counter impact in a certain time $$\tau _1$$ τ 1 such that the vibrations caused by the first impact are fully disappearing. In the current contribution this phenomenon is first studied for the simplest 1D bar vibration. The response function is introduced as a characteristic for such a phenomenon and, by studying its properties, it is possible to find both an impact duration time $$\tau $$ τ and an application time $$\tau _1$$ τ 1 for the counter impact leading to the wave cancellation. The result is generalized for any arbitrary homogeneous linear non-dissipative mechanical structure described by a semi-elliptic operator Lu . The counter impact can be determined in the same way as in the opposite direction. This general result is numerically illustrated for various operators Lu possessing relatively simple analytical solutions: for a simply supported and a clamped Bernoulli beam, for a fixed membrane and for a Kirchhoff plate. Three potential applications are discussed at the end: a set of verification examples for further analysis of time integration numerical schemes with the energy conservation property; straightforward transfer of cancellation conditions for the double impact to any convenient numerical method in mechanics, e.g. finite element method, iso-geometric method etc.; application of the result in engineering design of impacting devices (hammering etc.) in order to prevent recoil.

中文翻译：

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任意结构中有限持续时间双重冲击的波抵消条件

冲击系统中的共振现象可以定义为在周期性施加冲击期间振幅增加。消波现象被定义为应用一定条件使波完全消去。双冲击系统定义为施加一定持续时间 $$\tau $$ τ 的第一次冲击，然后在一定时间内施加反冲击 $$\tau _1$$ τ 1 使得振动引起受第一次冲击完全消失。在当前的贡献中，首先针对最简单的一维棒振动研究了这种现象。引入响应函数作为这种现象的特征，通过研究其特性，可以找到撞击持续时间 $$\tau $$ τ 和应用时间 $$\tau _1$$ τ 1导致波浪抵消的反作用。该结果适用于由半椭圆算子 Lu 描述的任意均匀线性非耗散机械结构。可以以与相反方向相同的方式确定反冲击。对于具有相对简单的解析解的不同算子 Lu 以数值方式说明了这个一般结果：对于简支和夹紧的伯努利梁，对于固定膜和基尔霍夫板。最后讨论了三个潜在的应用：一组验证实例，用于进一步分析具有能量守恒性质的时间积分数值方案；将双重冲击的抵消条件直接转移到力学中任何方便的数值方法，例如有限元方法、等几何方法等；