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Metamaterials with multiple resonators have been widely investigated for the purpose of generating multiple stop bands or broadening the attenuation bandwidth. The multiple resonators could be connected end to end in a line, namely, in-series connection, or connected individually to the host structures, namely, in-parallel connection. This paper investigates the influence of the resonator connection methodology on the frequency response functions of metamaterial beams with multiple resonators and exhibits an approach for optimizing their resonator distribution over the structure. The receptance functions of metamaterial beams with various resonator connection architectures are calculated by a transfer matrix model, which is verified through finite element model results. It is demonstrated that resonator interconnection architectures have a great impact on the global dynamic properties of metamaterials. An optimization strategy is subsequently proposed to find out the optimal resonator connection architectures and mass distributions that could minimize the maximal receptance functions in targeted single and multiple frequency ranges. The objective functions within single targeted frequency ranges are solved by the adoption of the genetic algorithm method. The weighted sum method is used to gain an optimal solution for multi-frequency range optimization. The metamaterial beams with optimal resonator connection methods and mass distributions demonstrate greatly enhanced vibration attenuation at frequencies of interest compared with other beams. The work is expected to provide the necessary theoretical basis and incentive for future researchers working on the design of metamaterials with extended, tuned, and optimized stop bands.

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This paper investigates the influences of nonperiodic rainbow resonators on the vibration attenuation of two-dimensional metamaterial plates. Rainbow metamaterial plates composed of thin host plates and nonperiodic stepped resonators are considered and compared with periodic metamaterial plates. The metamaterial plates are modelled with the finite element modelling method and verified by the plane wave expansion method. It was found that the rainbow metamaterial plates with spatially varying resonators possess broader vibration attenuation bands than the periodic metamaterial plate with the same host plates and total mass. The extension of attenuation bands was found not to be attributed to the extended bandgaps for the two-dimensional metamaterial plates, as is generally believed for a one-dimensional metamaterial beam. The complete local resonance bandgap of the metamaterial plates is separated to discrete bandgaps by the modes of nonperiodic resonators. Although the additional modes stop the formation of integrated bandgaps, the vibration of the plate is much smaller than that of resonators at these modal frequencies, the rainbow metamaterial plates could have a distinct vibration attenuation at these modal frequencies and achieve broader integrated attenuation bands as a result. The present paper could offer a new idea for the development of plate structures with broadband vibration attenuation by introducing non-periodicity.

The dynamics of tilting table behaves differently during five-axis machining due to the constant changes of the position of its center of mass which leads to different forces acting on parts of the transmission system. In this research, the lumped parameter method is used to model the dynamics of tilting table driven by worm and worm wheel in the tilting direction, where the varying stiffness of the transmission system at different tilting angles is considered. The impact testing experiments of tilting table system with tilting angles from 0° to 90° are also performed to verify the analytical model. The results from sensitivity analysis show that the three stiffnesses have a great effect on the variation of system natural frequency in the tilting direction, including the equivalent tangential meshing stiffness of worm and worm wheel, the torsional stiffness of worm wheel shaft, and the axial stiffness of worm supporting bearings. Moreover, the variations of system natural frequency with the three stiffnesses at different tilting angles are further investigated.