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Chi ama i libri sceglie Kobo e inMondadori. View Synopsis. Choose Store. Or, get it for Kobo Super Points! Skip this list. It is noteworthy that positive metadamping is affected by the pillar spacing more strongly, as D p decreases at a higher rate than D n with increasing a. Further insights by analytical model To further characterize the metadamping phenomenon, we resort to an analytical model of a locally resonant rod admitting only longitudinal motion which is the relevant mode of motion for all previous results.
A spring-mass oscillator is used to model the pillar. The extension of our analysis from the framework of Ref. Because we are interested in metadamping, we compare the dissipation levels of the locally resonant rod with those of a reference uniform rod. Unlike in Ref. Metadamping is a dissipation emergence phenomenon whereby the level of dissipation may be enhanced or reduced in a metamaterial compared to a statically equivalent material with the same mass and type and quantity of prescribed damping.
Both positive and negative metadamping have been demonstrated. Our results show that either an increase or a decrease in dissipation, beyond nominal levels, is realizable by the proposed concept.
Acoustic Metamaterials and Phononic Crystals
In both cases, metadamping may in principle be tailored to desired frequency ranges. While elastic metamaterials are known to provide strong spatial attenuation inside band gaps, this work demonstrates the ability to exhibit strong, or weak, temporal attenuation. Future work will extend the phenomenon of metadamping to the microscale and to waves driven at prescribed frequency.
The authors thank Professor Srikanth Phani for fruitful discussions. In order to solely focus on the effects of the pillar on dissipation and avoid effects stemming from the nature of the bonding, the pillars are not attached to the main beam by adhesion. Information on the FE implementation is available in Ref. Following the treatment of general damping models by the state-space approach, which was originally developed for vibrations of finite structures e.
The latter diagram provides us with the level of dissipation that each Bloch mode exhibits. Experimental frequency response functions FRF are obtained as follows. The beam under investigation is suspended using nylon cords in order to simulate free-free boundary conditions. Impulsive excitations are applied with an impact hammer PCB C02 at a point located at the center of the opposite cross-section, such that only the longitudinal modes are excited and measured. The measurements are collected with a NI-DAQ data acquisition system the frequency rate is set to As described in the main article, the experimental curve fitting of our numerical model is done in two steps.
In the second step, the model damping parameters are fine-tuned by correlating the finite and infinite problems using the proposed procedure demonstrated in Fig. This ensures accurate prediction of wavenumber-dependent damping ratios. Here we describe in the initial curve fitting step. The modal damping ratios associated with the two measured resonance frequencies in the range of interest kHz are extracted using the circle-fit modal analysis technique.
The code EasyMod [ 33 ] , which is available online, is employed for this purpose. Figure A1 compares the the numerical FRF and damping ratios with the corresponding experimental FRF and damping ratios using these damping parameters. As in the laboratory testing, an excitation is applied at the center of the cross-section on one end of the beam along the longitudinal axis and the displacement on the opposite end of the beam is recorded.
For the positive metadamping case, an initial displacement in the form of a Gaussian impulse is applied:. The frequency content of this profile falls mostly within the positive metadamping range of 2. For the negative metadamping case, a prescribed force is applied that is also of Gaussian form:. Here, the constants are selected such that most of the frequency content falls within the negative metadamping range of 3.
To study the effects of the pillar spacing on the degree of metadamping, we perform a parametric study and vary the unit-cell length a. Two metadamping metrics, D p and D n , are defined to help quantify the amount of positive and negative metadamping, respectively, in the unit cell. The procedure to obtain these metrics is as follows.
This is done by computing the longitudinal polarization for each Bloch mode using a method described in Ref. For the unpillared beam, we observe a single branch with dominant longitudinal motion; whereas for the pillared beam, two corresponding branches are extracted that exhibit longitudinal modes the transition from one branch to two branches is a manifestation of the resonance hybridization phenomenon that takes place due to the presence of the pillars.
We will refer to these two branches as the upper branch associated with positive metadamping and the lower branch associated with negative metadamping. As demonstrated in Fig. A2 , we define the positive [negative] metadamping metric as the area between the unpillared branch and the upper [lower] pillared branch, and between the left [right] and middle boundary lines. The middle line is drawn such that it passes through the extrema of the two pillared branches. The wavenumbers at which these extrema occur are determined based on the curvature i. In order to ensure accuracy in the area calculation of the two metadamping regions, linear interpolation is performed along the wavenumber axis on the unpillared and pillared branches, between the left and right boundary lines.
The areas are then calculated by slicing the two metadamping regions into quadrilaterals between consecutive wavenumbers and summing their areas. The formula used to find the area of each quadrilateral is as given by. The demonstration provide in Fig. The positive metadamping region is highlighted in blue and the negative metadamping one in black.
The final results of the parametric study are summarized in Fig. We want to ensure that the added mass of the pillar is not a factor into our metadamping study. Therefore, to show that the metadamping is due to the local resonance phenomena only and not due to the addition of extra mass that exhibits material damping, we perform a similar analysis as that of Fig.
The numerical time responses and their curve-fitted exponential functions are shown on Fig. In this section, we follow the recent work by Maznev [ 35 ] to develop an analytical characterization of positive and negative metadamping. To that end, we derive the dispersion relation for longitudinal motion in a thin homogeneous rod and in a rod with a spring-mass oscillator see Fig. The governing equation describing longitudinal displacements u in the rod is given by. The volume fraction of the oscillator is negligible compared to the rod; therefore, the density and the modulus of the rod are not affected.
The equation of motion of the spring-mass oscillator is. Note that the damping coefficient is chosen to be identical to the prescribed damping paramter loss factor for the supporting rod. From A9 , we arrive at the dispersion relation for a damped rod:. Combining A9 and A11 , we obtain the dispersion for the damped rod with an oscillator:. Rearranging A16 yields:. The left-hand side of A17 describes the interaction of the oscillator with the propagating waves in the rod, whereas the right-hand side describes the interaction of the two modes.
In the work presented in [ 35 ] , the author is interested in a so-called exceptional point , which can be described as the bifurcation point separating the strong-coupling from the weak-coupling regimes. A17 can be further simplified as:. According to [ 35 ] , the two modes coalesce at the exceptional point, requiring the square root term to be equal to zero. From Eq. Note that because of our choice of material damping model, i. In Fig. A5 B, we show the corresponding damping ratios. The damping branch associated with the acoustic mode lies above the damping ratio branch for the homogeneous rod for the entire wavenumber range positive metadamping.
Conversely, the damping branch associated with the optical mode lies below the damping ratio branch for the regular rod for the entire wavenumber range negative metadamping. In the first case Fig. Bacquet Ann and H. Hussein Ann and H. Abstract In civil, mechanical, and aerospace engineering, structural dynamics is commonly understood to be a discipline concerned with the analysis and characterization of the vibratory response of structures. A4 The middle line is drawn such that it passes through the extrema of the two pillared branches.
A17 The left-hand side of A17 describes the interaction of the oscillator with the propagating waves in the rod, whereas the right-hand side describes the interaction of the two modes. A19 According to [ 35 ] , the two modes coalesce at the exceptional point, requiring the square root term to be equal to zero.
Footnotes Dissipation engineering at the macroscopic level is additive; it builds on what may be achieved at the micrscopic level. Furthermore, it enables increased, or decreased, dissipation at very low frequencies, i. The MAC criterion is a measure of the degree of orthogonality between two vectors.
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Related Acoustic Metamaterials and Phononic Crystals: 173 (Springer Series in Solid-State Sciences)
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