Dynamics of Piezo-Embedded Negative Stiffness Mechanical Metamaterials: A Study on Electromechanical Bandgaps

Dynamics of periodic materials and structures have a profound historic background starting from Newton’s first effort to find sound propagation in the air to Rayleigh’s exploration of continuous periodic structures. This field of interest has received another surge from the early 21^st century. Elastic mechanical metamaterials are the exemplars of periodic structures that exhibit interesting frequency-dependent properties like negative Young’s modulus, negative mass and negative Poisson’s ratio in a certain frequency band due to additional feature of local resonance. This implies spatial periodicity of mechanical unit cells in engineered metamaterials exhibits properties that are beyond one can expect from conventional naturally occurring materials. Locally resonant units in the designed metamaterial facilitate bandgap formation virtually at any frequency for wavelengths much higher than lattice length of the unit. Whereas at higher frequencies for wavelengths equal to lattice size of the medium, the Bragg scattering phenomenon occurs, which also helps in the bandgap formation. Due to out of phase motion of multiple resonating units with lattice, there is a change in the dynamic behaviour (stiffness or mass) of the material as physical properties become frequency-dependent. Consequently, these extreme frequency-dependent physical properties modulate wave propagation through designed metamaterials and generate attenuation bands. In this research, we present the mathematical modelling of piezo-embedded negative stiffness metamaterials. For a finite number of units coupled equation of motion of the system is deduced using generalized Bloch theorem. Bloch theorem is used to solve periodic structure problems in various fields. Using this theory, the relationship between frequency and wave number can be derived and subsequently band structure of piezo-embedded negative stiffness metamaterial can be obtained. Successively, harvested power along with transmissibility is also computed for a chain of finite number of metamaterial units by using the backward substitution method. The addition of piezoelectric material at the resonating unit increases the damping and complexity of the solution. The results explicate that, insertion of piezoelectric material in the resonating unit provides better tunability of the system. By identifying critical parameters through an extensive non-dimensional study of this system, band structure of the designed metamaterial can be tailored. It enhances performance in terms of vibration attenuation and harvested energy. This research can be considered as first step towards designing the active elastic mechanical metamaterials.

The scope of this research is restricted to negative stiffness metamaterials. The geometry of each unit of the metamaterial is assumed to be less than the one-tenth of the wavelength of the propagated wave so that size of the unit neither interferes nor scatter the wave. Each unit of a piezo-embedded negative stiffness metamaterial vibrates with the harmonic excitation given to the system. On the basis of this, a brief mathematical model of the piezo-embedded negative mass metamaterial can be developed. Equilibrium of forces along X and Y directions will yield the equation of motion of the system. According to the energy harvesting circuit assumed, the equation of motion of the piezo-embedded resonating unit due to electromechanical coupling can be written. Now all these equations can be solved simultaneously for finding displacement and voltage produced by piezoelectric material in each unit. The dispersion relationship can be established using Bloch theorem and subsequently band structure of the system can be plotted by varying different system parameters. The limiting frequencies corresponding to the first attenuation band can be identified. Based on this, the lower and higher parts of the first attenuation band will be identified as LA1 and HA1 respectively. These two sub-bands can be divided about unity on the frequency axis. Hence, LA1 and HA1 are critical for the maximization of the first attenuation band. For wideband vibration control and energy harvesting, reduction of parameters and the identification of the optimal combination of the critical parameters are quintessential.  This provides motivation for a comprehensive study of the system to elucidate how the width of the first attenuation band gets affected by the identified non-dimensional system parameters.