Introduction
Research motivation
- What is metamaterial?
- Why low-frequency and broadband flexural metamaterial?
- Limitation of previous studies
By doing this, we can easily control the wave propagation speed and significantly reduce the operating frequency of the metamaterial. Although the need of the low-frequency broadband is bandgap, it is difficult to achieve the bandgap in low-frequency and broadband at the same time.
Research objective
However, it also has a drawback; it requires very large periodicity and heavy mass to reach the band gap in a low frequency range which severely impairs the practical use of metamaterials. However, they still rely on additional conditions or materials that impose a new limitation that inhibits the practical use of metamaterials.
Paper organization
The stiffness ratio and inertia ratio of the unit cell are plotted as shown in Figure B-6(a). Proposed unit cell dispersion curve (b) for negative group velocity and (e) positive group velocity.
Background Theory for flexural metamaterials
Equivalent mass-spring system
The advantage of the corresponding mass spring system is that it is intuitive and practical. To address this point, we convert the metamaterial in a continuum structure to the corresponding mass-spring system, which has.
Introducing extended mass-spring system for flexural wave
This allows the wave propagating through the metamaterial to be easily analyzed via a mass-spring system for the frequency range of interest. An equivalent mass-spring system is proposed for a flexural wave propagating through a one-dimensional Timoshenko beam, as shown in Fig. 2-2(a).
Theoretic analysis of extended mass-spring
The proposed unit cell, and (b) the dispersion curve of the unit cell in three dimensions. It is clear from the shape of the scatter curve that this is clearly visible in Figure C-2.
Vibration suppression in broadband low-frequency based on flexural metamaterials
Introduction
In the previous section, we propose the extended mass-source system for the theoretical analysis of flexural wave in metamaterials. As a result, we show the possibility of achieving the desired bandwidth by adjusting the parameters of the extended mass-spring system.
In this section, the condition for wide band gap at low frequency will be discussed and the bending metamaterial for wide band gap at low frequency is designed and validated. mass m is required for higher band-closing frequency w2 and lower bending stiffness b and higher rotational inertia I are required for lower band-opening frequency w1.
3-2 Distribution of strain and transverse moment of a beam under (a) pure bending moment and (b) pure shear stress. In the case of the spring part, high shear stiffness and low bending stiffness must be achieved at the same time to achieve a low frequency broadband.
Numerical validation
The error between the analytical and numerical solutions at a high frequency range occurs due to the limitation of the mass-spring system as mentioned in chapter 2. However, in this study we are interested in the bandgap frequency range where the extended mass -spring system holds well. 3-5 (a) Numerical simulation setup for transmission test in the finite structure. b) Transmission curve of the finite structure in (a).
As a second step, the transmission curve of the metamaterial composed of a finite number of unit cells is considered.
Experimental validation
Then the signal is amplified by the amplifier (Power amplifier type 2718, B&K) and the signal is transferred to the shaker to activate the metamaterial. After activation, the z-direction velocity at the top surface of mass #1 and mass #4 is measured by the laser doppler vibrometer (CLV-2534, Polytec). Then, the transmission curve is obtained from the velocity amplitude ratio of mass #1, and mass #4 is plotted as a red circle in Figure 3-6(b).
Based on these three steps of validation, we validated our analytical model, extended mass spring system and unit cell design for the low-frequency broadband band gap by numerical simulation and experiment.
- change thickness of mass and spring
- change shape of mass and spring
In addition, the optical branch of the bending wave is also shown in Figure B-2 as a solid line. In the previous section, we investigated the condition for determining the group velocity of the optical branch of the bending wave. The group velocity of the optical branch agrees with the prediction from the conditions in Eq.
A positive group velocity unit cell is designed as shown in Figure B-7(d). B-28) is satisfied, which means that the group velocity of the optical branch has a.
Highly tunable metamaterial cavity for elastic energy harvesting
Introduction
Therefore, tuning the frequency of the cavity mode is quite simple, by changing the weight or stiffness of the cavity. In addition, the wide bandgap not only provides high frequency tunability, but also provides strong confinement for the cavity mode. Therefore, the vibrational energy can be strongly localized within the cavity due to the wide band gap [52].
In addition, another problem in the metamaterial cavity is that the performance of the cavity mode can be lower than the normal beam's resonance.
Metamaterial cavity design
When a wave with a frequency band is incident on a metamaterial, the wave cannot exist in the form of a propagating wave. Instead, it can exist in the form of an evanescent wave, which is a decaying wave as it expands and disappears. But when the evanescent wave reaches the cavity where the wave can exist as a propagating wave, it turns into a propagating wave and forms a standing wave in the cavity, i.e. is localized within the cavity.
They show good matching and a wide bandgap in the low-frequency region is observed.
Metamaterial cavity design: tunability in frequency
Based on the metamaterial design, the cavity mode is achieved by placing the cavity as shown in Figure 4-3(a). The numerical simulation is performed to confirm the cavity mode in the dispersion curve. The passband is blurred by a blue box, and the cavity mode is plotted in the colored line.
These numerical results show that the cavity mode frequency can be tuned by changing the length of the cavity based on the wide bandgap in the low-frequency range.
Metamaterial cavity design: tunability in performance
Amplification ratio with respect to frequency and side beam length. It can be seen that the cross point of these two lines indicates the highest amplification of the cavity mode in Figure 4-5 (b). It means that the cavity mode amplification performance can be improved by adjusting the length of the side beam.
The cavity mode enhancement shows significantly better performance than the metamaterial without the side beam in Figure 4-4(b).
Experimental validation
B-13) has a negative value, from which we can directly conclude that the group velocity of the optical branch has a negative value for IBZ. Physical interpretation of the group velocity of the optical branch. a) Bending wave dispersion curve for (a) negative group velocity, (b) positive group velocity, and (c) positive-negative group case. In the case of a positive group velocity, the origin of the optical branch changes to the 2nd-order mode, as shown in Figure B-4(b).
Therefore, we analyze the behavior of the optical branch of the bending wave, especially about the group velocity. Violation of the mass spring system assumption in the parametric study. a) the parameter map of the unit cell. The four-dispersion curve for the unit cell with the extreme parameters is plotted as shown in Figure D-1(b).
Conclusion
Theoretical analysis on the two-dimensional mass-spring system
Numerical validation on the two-dimensional mass-spring system
To validate the analytical predictions in the previous section, the eigenfrequency simulation is performed for the unit cell in Figure A-2(a). The numerically calculated gap appears from 1597 Hz to 2736 Hz which coincides with the analytical band. From these results, our two-dimensional mass-stretching system is valid for analysis for two-dimensional bending metamaterial.
Analytic investigation on the group velocity of the optical branch
Analytic investigation for the group velocity of optical branch
Therefore, only the numerator is related to the sign of the group velocity of the optical branch. B-10) can be extended as shown below. B-11). For the given wavenumber, if the optical branching of the bending wave above the curve given by Eq. B-14. This means that the optic branch has both positive group velocity and negative group velocity in IBZ.
In summary, there are three cases for the optical branch: having a negative, positive, and positive-negative group velocity.
Physical explanation for the group velocity of optical branch
This reveals the reason why the optical branch has a positive group velocity, since it originates in the higher order mode. The cutoff frequency of the 2nd mode appears earlier than the folded 1st-order mode emerges from the imaginary part, i.e., before the Bragg gap closes. In the case where the optical branch has positive-negative group velocities, the Bragg gap closing frequency and the cutoff frequency may compete with each other.
Consequently, the bending wave has several signs of the group velocity is that the existing mode of the optical branch is changed as the stiffness ratio and the inertia ratio change.
Numerical validation
The stiffness ratio of the unit cell is controlled by changing the distance between the two connections. Thus, the mode shape of the unit cell with various combinations of the thicknesses of the cylinder and spring is observed as shown in Figure D-1. The parameter map shows the range of the spring thickness and cylinder thickness of the unit cell in Figure 3-7(a).
Thus, the thickness of the cylinder varies from 10 mm to 20 mm as shown in Figure D-1(a). The area marked with a red box shows the possible range of the unit cell dimension. The reason for this error can be seen in the shape of the unit cell mode at the cutoff frequency where k=0, as shown in Figure D-1(c).
Property of the unit cell for other types of waves
Violation of the assumption for mass-spring system in the parametric study
Amplitude-induced bandgap: A new type of bandgap for nonlinear elastic metamaterials. Journal of the Mechanics and Physics of Solids. Hybrid elastic metamaterial with negative mass density and adjustable bending stiffness. Journal of the Mechanics and Physics of Solids. Elastic wave propagation in elastic metamaterials containing parallel multiresonators. Journal of Physics D: Applied Physics.
Energy capture below the wavelength of elastic waves in a metamaterial. The Journal of the Acoustical Society of America, 136(2), EL192-EL198.
