Chapter 21 Analysis of Traveling Wave Properties of Mechanical Metamaterial Structures: Simulation and Experiment Hannes Fischer and Sebastian Tatzko Abstrac t The steady-state response of harmonically excited structures can exhibit a significant traveling wave ratio. Local excitation of structures with locally increased damping or even structures that are proportionally damped, for example, lead to wave propagation phenomena. Since damping distribution plays a key role in formation of traveling waves, it needs to be considered in the dynamic analysis. In this chapter, we analyze the steady-state vibration behavior of 3D printed metamaterial structures. The investigated parts are made of resin and steel by laser sintering. The dynamic analysis with special attention to traveling wave effects is simulated based on finite element method and experimentally validated. Due to the complex geometry of the metamaterial structure, fine meshing is necessary for accurate results, making reduction techniques inevitable. A combination of modal reduction and dynamic condensation is used to obtain the simulated results. In the laboratory, laser scanning vibrometry is used to measure the entire structure and validate the simulations. We show in both simulation and experiment that the studied structures exhibit both standing waves with locally fixed nodal lines and traveling nodal lines with significant traveling wave content, depending on the excitation frequency. Keyword s Local damping · Modal damping · Traveling waves · Dynamic condensation · Metamaterial 21.1 Introduction Harmonic excitation on a mechanical structure will create structural waves [1]. Typically, these waves are reflected at structural boundaries, and superposition of waves leads to a standing wave pattern. However, these standing waves, which at resonance are close to eigenmodes of the structure, are only a special solution to the general wave form of vibrations. Especially, with non-negligible damping, the reflected wave will not match the original wave in amplitude resulting in a significant traveling wave portion. Formation of traveling waves is not restricted to a certain type of damping, but the combination of excitation and damping is crucial. Even proportional damping that is often used due to its diagonalization property leads to traveling waves when the excitation is applied by a local force [2]. Furthermore, local damping will generate energy transfer from excitation source to the damper with a characteristic phase shift in the steady-state vibration response. Complex phenomena associated with waves can already be observed in two degree of freedom mechanical systems [3] and vibrating strings with just one local dissipative attachment [4]. Indeed, traveling wave motion has been used for some time in technical applications, such as ultrasonic motors [5] or granular transportation [6]. Today, the researchers are concerned with wave propagation and its manipulation to create waveguides and bandgaps passively through metamaterial structures [7]. Thus, the fundamental mechanical theories on periodic waves combined with efficient numerical methods for model reduction and substructuring are utilized to design and analyze structures with specific properties [8]. H. Fischer ( ) · S. Tatzko Institute of Dynamics and Vibration Research, Department of Mechanical Engineering, Leibniz University Hannover, Garbsen, Germany e-mail: fischer@ids.uni-hannover.de; tatzko@ids.uni-hannover.de © The Society for Experimental Mechanics, Inc. 2024 B. J. Dilworth et al. (eds.), Topics in Modal Analysis & Parameter Identification, Volume 9 , Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-34942-3_21 169
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