72 H. Wo¨hler and S. Tatzko Modeling of Metamaterial Structure This work is based on a metamaterial model consisting of a lumped-mass system, which is described via mass matrix M, damping matrixCand stiffness matrixK. In this structure, the resonator effect is utilized to achieve a bandgap considering the displacement amplitudes from one end to the other. Figure 1 shows that seven resonators are coupled to the central part of the host structure which is an oscillator chain with 15 degrees of freedom (DOFs). Each host mass is coupled to the environment via a linear spring. The whole metamaterial structure is base excited via stiffness at the first DOFxexc. a) kr k k k k m mr m m m m b) k k k k k kbase u(t) xexc xend Fig. 1 Lumped-mass model of the metamaterial structure with base excitation: a) full system with 15 host masses and seven resonators; b) detailed view of parts with labeling. Table 1 shows the parameters of the lumped-mass model of the metamaterial structure in Figure 1. The resonator masses of the reference model are 10%of the host masses mr = 0.1· mfor which a significant bandgap is already observed. For comparison, the resonator masses are doubled in another variant to 20 %of the host masses mr = 0.2 · mto show the effect of bandgap enlargement by simply adding mass. Figure 1 does not show any dashpot dampers, but they are modeled proportional to the stiffnesses k and kr using Rayleigh-damping in the form of C=βK. Table 1 Parameters of lumped-mass model and equally tuned resonators, cf. Figure 1. kbase in N m minkg c in Ns m k in N m mr inkg cr in Ns m ωr in rad s kr in N m equal (reference) 1000 0.1 1.5· 10−7 8000 0.01 15· 10−7 440 1936 equal 2-mass 1000 0.1 1.5· 10−7 8000 0.02 15· 10−7 440 3872 The focus is on investigating the influence of different resonator detuning patterns with respect to the bandgap width at DOFxend. Figure 2 shows the different detuning patterns as bar plots. The metamaterial structure with seven resonators of equal mass each tuned via stiffnesses kr to the natural frequency of ωr,1 −7 = 440 rad s serves as a reference, see Figure 2 a). Two different detuning patterns with five detuning levels each are investigated for the seven resonators. On the one hand, an alternating ABABABA-pattern, in which the resonators are detuned by ±5%(±10%, ±15%, ±20%, ±25%) in relation to the base frequency of 440 rad s . These are referred to as AB-5, AB-10, AB-15, AB-20 and AB-25, see Figure 2 b)-f). On the other hand, a triangular Tri-tuning in the form of ABCDCBA is investigated. Here, resonators #1 and #7 are detuned by −5%(−10%, −15%, −20%, −25%) and resonator #4 by +5%(+10%, +15%, +20%, +25%). The resonators in between are detuned according to linear interpolation. The latter variants are referred to as Tri-5, Tri-10, Tri15, Tri-20 and Tri-25, see Figure 2 h) - l). Note that the resonator masses for all configurations of patterns AB-tuning and Tri-tuning are equal to the resonator mass of the reference structure. This way the resonator masses stay unchanged and the bandgap width is only affected by detuning and not by mass effect. The eigenfrequency is thus only set via the resonator stiffnesses kr. Analysis of Bandgap Properties The analysis of the band gap properties with regard to the band gap width is carried out in two ways, which are explained in the following section. First, the bandgap width will be determined by modeling the metamaterial structure as a unit cell
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