Modal analysis and demonstration of metastructure combining local resonators and an autonomous synchronized switch damping circuit

A locally resonant metastructure is a promising approach for low-frequency vibration attenuation, whereas the attachment of many resonators results in unnecessary and multiple resonance outside the bandgap. To address this issue, we propose a damping metastructure combining local resonators and an autonomous synchronized switch damping circuit. On the basis of modal analysis, we derive an electromechanically coupled equation of the proposed metastructure. The piezo ceramics, which are attached on a small portion of the metastructure and connected to the circuit, remarkably decrease the magnitude of the resonant vibration with no extra sensors, signal processors, or power sources. The displacement at unnecessary resonance was decreased by approximately 75%. The results of the coupled analysis were similar to the experimentally observed results in terms of the location and width of the bandgap on the frequency axis and the decreased displacement for the circuit. The proposed technique can overcome the disadvantage of the metastructure.

Characters of Explicit Solutions for a Semidiscrete Integrable Coupled Equation

A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.

Two Isospectral-Nonisospectral Super-Integrable Hierarchies and Related Invariant Solutions

In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.

N-fold Darboux transformation of a six-field integrable lattice system

In this paper, we studied a semidiscrete coupled equation, which is integrable in the sense of admitting Lax representations. Proposed first by Vakhnenko in 2006, local conservation laws and one-fold Darboux transformation were presented with different forms, respectively, in O. O. Vakhnenko, J. Phys. Soc. Jpn. 84, 014003 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015). On the basis of these results, we principally construct [Formula: see text]-fold Darboux transformation by means of researching gauge transformation of its Lax pair, and work out its explicit multisolutions. Given a set of seed solutions and appropriate parameters, we can calculate two-soliton solutions and plot their figures when [Formula: see text].

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 21st 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 specific frequency band due to additional feature of local resonance. In this research, we present the modeling of piezo-embedded negative stiffness metamaterials by considering a shunted inductor energy harvesting circuit. For a chain of a finite number of metamaterial units, the coupled equation of motion of the system is deduced using generalized Bloch’s theorem. Successively, the backward substitution method is applied to compute harvested power and the transmissibility of the system. Additionally, through the extensive non-dimensional study of this system, the proposed metamaterial band structure is investigated to perceive locally resonant mechanical and electromechanical bandgaps. The results explicate that the insertion of the piezoelectric material in the resonating unit provides better tun-ability for vibration attenuation and harvested energy.

Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg-de Vries equation

Two nonlocal versions of the semi-discrete modified Korteweg-de Vries equation are derived by different nonlocal reductions from a coupled equation set in the Ablowitz–Ladik hierarchy. Different kinds of exact solutions in terms of double Casoratians to the reduced equations are obtained by imposing constraint conditions on the double Casorati determinant solutions of the coupled equation set. Dynamics of the soliton solutions for the real and complex nonlocal semi-discrete modified Korteweg-de Vries equations are analyzed and illustrated by asymptotic analysis.

Soliton solutions of coupled higgs field equation via the trial equation method

Since a few recent decades, investigation of nonlinear evolution equations (NLEEs) is becoming an important area of research as they have a variety of applications in various branches of social and scientific disciplines like Ecology, Social Dynamics, Financial Mathematics, Engineering and many branches of Physics such as Biophysics, Chemical Physics, Fibre Optics, Fluid Mechanics, Neuro-physics, Particle Physics, Solid State Physics and many more. Many powerful and efficient methods of finding exact solutions of NLEEs have been proposed so far and the Trial Equation Method [ 1 - 5] is one of them. Many authors have successfully used the method in finding exact solutions of a number of NLEEs. In the present paper, soliton solutions of the Coupled Higgs Field Equation [ 6 - 10 ] are being obtained using the Trial Equation Method. The Coupled Higgs Field Equation describes system of conserved scalar nucleons interacting with neutral scalar mesons in particle physics. This coupled equation has applications in the studies of Field Theory and Electromagnetic waves as well. This coupled equation introduces the Higgs field to illustrate the mechanism of generation of mass for Gauge Bosons. The Coupled Higgs Field Equation is generally expressed as the following pair of NLEEs (3) and (2) Here, x and t are spatial and temporal variables respectively, the function is a complex scalar nucleon field, the function is a real scalar meson field, are arbitrary real constants and the subscripts denote partial differentiations with respect to them.Using the Trial Equation Method, the above coupled NLEE is to be solved to obtain some soliton solutions.