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Determination of effective mass density and modulus for resonant metamaterials
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View: Figures


Image of FIG. 1.
FIG. 1.

Longitudinally vibrating rods (a) constrained by distributed springs, (b) with continuous, and (c) discrete spring and mass (oscillator) attachments. (d) Excitation of the rod at one end with the mass attached at the other end.

Image of FIG. 2.
FIG. 2.

(Color online) Calculated transfer functions for rods constrained using different methods. The vibration response was reduced at low frequencies when the rod was constrained by springs connected to a fixed floor. When constrained by continuous or discrete oscillators, the rod displacement response was significantly reduced near the oscillator natural frequency of 580 Hz.

Image of FIG. 3.
FIG. 3.

(Color online) Calculated (a) effective modulus and (b) mass density from predicted transfer functions of rods constrained by different methods. The passive oscillators considered in this study exhibited minimal influence on the effective modulus. The effective mass density was complex-numbered when the spring element of the oscillators was viscoelastic, and the real part became negative near the oscillator natural frequency.

Image of FIG. 4.
FIG. 4.

(Color online) Predicted effective mass density for the configuration in Fig. 1(c) of different loss factors of the attached discrete oscillators. Magnitude of the positive and negative maximums near the natural frequency of the oscillators decreased due to damping in the oscillators.

Image of FIG. 5.
FIG. 5.

(Color online) Reflection and transmission coefficients predicted for rods. Transmission of vibration energy was prohibited from oscillator attachments.

Image of FIG. 6.
FIG. 6.

(Color online) Experimental setup to measure the dynamic properties of longitudinally vibrating rods attached with multiple oscillators.

Image of FIG. 7.
FIG. 7.

(Color online) Measured transfer functions for rods with and without discrete oscillator attachments. The natural frequency decreased especially for the first mode from the influence of the oscillator. The measured values showed good agreement with the predictions shown in Fig. 2.

Image of FIG. 8.
FIG. 8.

(Color online) (a) Effective modulus and (b) mass density obtained from the measured transfer functions for vibrating rods with oscillator attachments. As seen from the predicted results, the oscillator attachments showed minimal influence on the effective modulus. The effective mass density became negative when the transmission coefficient shown in Fig. 5 was small, near the oscillator natural frequency.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Determination of effective mass density and modulus for resonant metamaterials