Abstract
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 and then the application of a counter impact in a certain time tau _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 and an application time tau _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.
Highlights
The structural resonance phenomenon is well known as an increase of the vibration amplitude if the frequency of the exciting force is approaching the eigenfrequency of the structure
The first impact of a final duration τ is applied at a certain point and is causing the first wave, the second counter impact is applied in a certain time τ1 such that the vibrations are fully cancelled
A response function is introduced in order to calculate all parameters of the double impact leading to the wave cancellation conditions
Summary
The structural resonance phenomenon is well known as an increase of the vibration amplitude if the frequency of the exciting force is approaching the eigenfrequency of the structure. Yang [36] consider vibration of simple beams due to trains, in which moving loads have been represented via compositions of Dirac delta functions and Heaviside functions Both wave cancellation and resonance conditions have been. Properties of the response function allows to construct the second impact satisfying the wave cancellation condition for any arbitrary mechanical structure subjected to the double impact problem. 4. Since the general result is applicable for a large number of mechanical problems—e.g., various beam and shell models for engineering structures, wave propagation in 2D and 3D continuum etc.—the double impact is shown selectively for models possessing relatively simple analytical solutions: – a supported Bernoulli beam; – a cantilever Bernoulli beam; – a membrane; – a Kirchhoff beam
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