Hamiltonian long wave expansions for internal waves over a periodically varying bottom

We derive a Hamiltonian formulation for two-dimensional nonlinear long waves between two bodies of immiscible fluid with a periodic bottom. From the formulation and using the Hamiltonian perturbation theory, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves and unidirectional equations that are similar to the KdV equation for the case in which the bottom possesses short length scale. The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators.     

Nonlinear hardening and softening resonances in micromechanical cantilever-nanotube systems originated from nanoscale geometric nonlinearities

Micro/nanomechanical resonators often exhibit nonlinear behaviors due to their small size and their ease to realize relatively large amplitude oscillation. In this work, we design a nonlinear micromechanical cantilever system with intentionally integrated geometric nonlinearity realized through a nanotube coupling. Multiple scales analysis was applied to study the nonlinear dynamics which was compared favorably with experimental results. The geometrically positioned nanotube introduced nonlinearity efficiently into the otherwise linear micromechanical cantilever oscillator, evident from the acquired responses showing the representative hysteresis loop of a nonlinear dynamic system. It was further shown that a small change in the geometry parameters of the system produced a complete transition of the nonlinear behavior from hardening to softening resonance.     

Nonlocal Galerkin Strip Transfer Function Method for Vibration of Double-Layered Graphene Mass Sensor

Double-layered graphene sheets(DLGSs) can be applied to the development of a new generation of nanomechanical sensors due to their unique physical properties. A rectangular DLGS with a nanoparticle randomly located in the upper sheet is modeled as two nonlocal Kirchhoff plates connected by van der Waals forces. The Galerkin strip transfer function method which is a semi-analytical method is developed to compute the natural frequencies of the massplate vibrating system. It can give exact closed-form solutions along the longitudinal direction of the strip. The results obtained from the semi-analytical method are compared with the previous ones, and the differences between the single-layered graphene sheet(SLGS) and the DLGS mass sensors are also investigated. The results demonstrate the similarity of the in-phase mode between the SLGS and DLGS mass sensors. The sensitivity of the DLGS mass sensor can be increased by decreasing the nonlocal parameter, moving the attached nanoparticle closer to the DLGS center and making the DLGS smaller. These conclusions are helpful for the design and application of graphene-sheet-based resonators as nano-mass sensors.     

Noether symmetry method for Birkhoffian systems in terms of generalized fractional operators

Compared with the Hamiltonian mechanics and the Lagrangian mechanics, the Birkhoffian mechanics is more general. The Birkhoffian mechanics is discussed on the basis of the generalized fractional operators, which are proposed recently. Therefore, differential equations of motion within generalized fractional operators are established. Then, in order to find the solutions to the differential equations, Noether symmetry, conserved quantity, perturbation to Noether symmetry and adiabatic invariant are investigated. In the end, two applications are given to illustrate the methods and results.     

Nanomechanical Unfolding of Self-Folded Graphene on Flat Substrate

Experimental Mechanics - We report the nanomechanical unfolding of individual self-folded graphene flakes on a flat substrate by using atomic force microscopy techniques. The nanomechanical...     

Modeling and analysis of electrostatic MEMS filters

We present an analytical model and closed-form expressions describing the response of a tunable MEMS filter made of two electrostatic resonators coupled by a weak microbeam. The model accounts for the filter geometric and electric nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system to produce a reduced-order model. We predict the filter deflection and static pull-in voltage by solving a boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the filter poles (the natural frequencies delineating the filter bandwidth). We found a good agreement between the results obtained using our model and published experimental results. We found that, when the input and output resonators are mismatched, the first mode is localized in the softer resonator whereas the second mode is localized in the stiffer resonator. We demonstrated that mismatch between the resonators can be countermanded by applying different DC voltages to the resonators. As the effective nonlinearities of the filter grow, multi-valued responses appear and distort the filter performance. Once again, we found that the filter can be tuned to operate linearly by choosing a DC voltage that makes the effective nonlinearities vanish.     

A Whitham–Boussinesq long-wave model for variable topography

We study the propagation of water waves in a channel of variable depth using the long-wave asymptotic regime. We use the Hamiltonian formulation of the problem in which the non-local Dirichlet–Neumann operator appears explicitly in the Hamiltonian, and propose a Hamiltonian model for bidirectional wave propagation in shallow water that involves pseudo-differential operators that simplify the variable-depth Dirichlet–Neumann operator. The model generalizes the Boussinesq system, as it includes the exact dispersion relation in the case of constant depth. Analogous models were proposed by Whitham for unidirectional wave propagation. We first present results for the normal modes and eigenfrequencies of the linearized problem. We see that variable depth introduces effects such as a steepening of the normal modes with the increase in depth variation, and a modulation of the normal mode amplitude. Numerical integration also suggests that the constant depth nonlocal Boussinesq model can capture qualitative features of the evolution obtained with higher order approximations of the Dirichlet–Neumann operator. In the case of variable depth we observe that wave-crests have variable speeds that depend on the depth. We also study the evolutions of Stokes waves initial conditions and observe certain oscillations in width of the crest and also some interesting textures and details in the evolution of wave-crests during the passage over obstacles.     

Periodic Orbits in Hamiltonian Systems with Involutory Symmetries

We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetry. In both classes, the involution reverses the sign of the Hamiltonian function, and the system is in 1:1 resonance. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R, and furthermore that there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian (correcting a minor error in their paper). In the second problem we study an equivariant Hamiltonian system with a symmetry S that acts anti-symplectically. Generically, there is no S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we prove the existence of at least 2 and at most 12 families of non-symmetric periodic solutions. We conclude with a brief study of systems with both forms of symmetry, showing they have very similar structure to the system with symmetry R.     

The nanogranular nature of C-S-H

Despite its ubiquitous presence as binding phase in all cementitious materials, the mechanical behavior of calcium-silicate-hydrates (C-S-H) is still an enigma that has deceived many decoding attempts from experimental and theoretical sides. In this paper, we propose and validate a new technique and experimental protocol to rationally assess the nanomechanical behavior of C-S-H based on a statistical analysis of hundreds of nanoindentation tests. By means of this grid indentation technique we identify in situ two structurally distinct but compositionally similar C-S-H phases heretofore hypothesized to exist as low density (LD) C-S-H and high density (HD) C-S-H, or outer and inner products. The main finding of this paper is that both phases exhibit a unique nanogranular behavior which is driven by particle-to-particle contact forces rather than by mineral properties. We argue that this nanomechanical blueprint of material invariant behavior of C-S-H is a consequence of the hydration reactions during which precipitating C-S-H nanoparticles percolate generating contact surfaces. As hydration proceeds, these nanoparticles pack closer to center on-average around two characteristic limit packing densities, the random packing limit (η=64%) and the ordered face-centered cubic (fcc) or hexagonal close-packed (hcp) packing limit (η=74%), forming a characteristic LD C-S-H and HD C-S-H phase.     

The static and dynamic behavior of MEMS arch resonators near veering and the impact of initial shapes

We investigate experimentally and analytically the effect of initial shapes, arc and cosine wave, on the static and dynamic behavior of microelectromechanical systems (MEMS) arch resonators. We show that by carefully choosing the geometrical parameters and the initial shape of the arch, the veering phenomenon (avoided-crossing) among the first two symmetric modes can be strongly activated. To demonstrate this, we study electrothermally tuned and electrostatically driven initially curved MEMS resonators. Upon changing the electrothermal voltage, we demonstrate high frequency tunability of arc resonators compared to the cosine-configuration resonators for the first and third resonance frequencies. For arc beams, we show that the first resonance frequency increases up to twice its fundamental value and the third resonance frequency decreases until getting very close to the first resonance frequency triggering the veering phenomenon. Around the veering regime, we study experimentally and analytically the dynamic behavior of the arc beam for different electrostatic loads. The analytical study is based on a reduced order model of a nonlinear Euler–Bernoulli shallow arch beam model. The veering phenomenon is also confirmed through a finite-element multi-physics and nonlinear model.     

Completeness of system of root vectors of upper triangular infinite-dimensional Hamiltonian operators appearing in elasticity theory

This paper deals with a class of upper triangular infinite-dimensional Hamiltonian operators appearing in the elasticity theory.The geometric multiplicity and algebraic index of the eigenvalue are investigated.Furthermore,the algebraic multiplicity of the eigenvalue is obtained.Based on these properties,the concrete completeness formulation of the system of eigenvectors or root vectors of the Hamiltonian operator is proposed.It is shown that the completeness is determined by the system of eigenvectors of the operator entries.Finally,the applications of the results to some problems in the elasticity theory are presented.     

Mei symmetries and conserved quantities for non-conservative Hamiltonian difference systems with irregular lattices

Noether conserved quantities and Mei symmetries for non-conservative Hamiltonian difference systems with irregular lattices are studied. The generalized Hamiltonian equations of the systems are given on the basis of the transformation operators in the space of discrete Hamiltonians. The Lie point transformations acting on the lattice, as well as the difference equations, and the determining equations of Mei symmetries are obtained for the systems. The discrete versions of Noether conserved quantity are constructed by the Mei symmetries. An example is presented to illustrate the results.     

Hamiltonian long-wave approximations to the water-wave problem

This paper presents a Hamiltonian formulation of the water-wave problem in which the non-local Dirichlet-Neumann operator appears explicitly in the Hamiltonian. The principal long-wave approximations for water waves are derived by the systematic approximation of the Dirichlet-Neumann operator by a sequence of differential operators obtained from a convergent Taylor expansion of the Dirichlet-Neumann operator. A simple and satisfactory method of obtaining the classical two-dimensional approximations such as the shallow-water, Boussinesq and KdV equations emerges from the process. A straightforward transformation theory describes the relationship between the classical symplectic structure appearing in the water-wave problem and the various non-classical symplectic structures that arise in long-wave approximations. The discussion extends to include three-dimensional approximations, including the KP equation.     

HAMILTON OPERATORS AND HOMOTHETIC MOTIONS IN R3

IntroductionPfaff[1]hasbeendefinedaquaternionproductonleaf (theplanecontainingtheOx_axis) .Byaidoftheproduct,anewproductisdefinedonplaneswhichpassesthroughtheorigindonotcontaintheOx_axisandsomeofthepropertiesofthisproducthavebeeninvestigated .Agrawal[2 ]gavesomealgebricpropertiesofHamiltonoperators.Also ,quaternionswereexpressedintermsof 4× 4matricesbymeansoftheseoperators.Yayli[3]gavehomotheticmotionswithaidoftheHamiltonoperatorsatE4 .Hacisalihoglu[4 ]showedthatallHomotheticmotionsinn_di…     

Decomposition methods for time‐domain Maxwell's equations

Decomposition methods based on split operators are proposed for numerical integration of the time‐domain Maxwell's equations for the first time. The methods are obtained by splitting the Hamiltonian function of Maxwell's equations into two analytically computable exponential sub‐propagators in the time direction based on different order decomposition methods, and then the equations are evaluated in the spatial direction by the staggered fourth‐order finite‐difference approximations. The stability and numerical dispersion analysis for different order decomposition methods are also presented. The theoretical predictions are confirmed by our numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

基因芯片纳米挠度和表面应力的解析预测

本文利用能量法解析分析了无标记生物检测中基因芯片的纳米力学行为.首先,考虑微悬臂梁机械能和基因层静电能、水合能和构型熵,建立了基因芯片能量模型,并采用泰勒级数展开法.获得了能量势的一阶近似式.其次,利用能量最小原理,得到了芯片稳态弯曲的曲率半径、纳米挠度和表面应力的解析表达式,从而克服了求解多极值能量泛函数值解的困难.最后,预测了DNA的链长和种植密度对芯片纳米力学行为的影响,同时将表面应力的解析预测结果与有关实验数据进行了比较,证明了本文方法的可靠性.     

Nanoscale Broadband Viscoelastic Spectroscopy of Soft Materials Using Iterative Control

A novel approach to nanoscale broadband viscoelastic spectroscopy is presented. The proposed approach utilizes the recently developed modeling-free inversion-based iterative control (MIIC) technique to achieve accurate measurement of the material response to the applied excitation force over a broad frequency band. Scanning probe microscope (SPM) and nanoindenter have become enabling tools to quantitatively measure the mechanical properties of a wide variety of materials at nanoscale. Current nanomechanical measurement, however, is limited by the slow measurement speed: the nanomechanical measurement is slow and narrow-banded and thus not capable of measuring rate-dependent phenomena of materials. As a result, large measurement (temporal) errors are generated when material is undergoing dynamic evolution during the measurement. The low-speed operation of SPM is due to the inability of current approaches to (1) rapidly excite the broadband nanomechanical behavior of materials, and (2) compensate for the convolution of the hardware adverse effects with the material response during high-speed measurements. These adverse effects include the hysteresis of the piezo actuator (used to position the probe relative to the sample); the vibrational dynamics of the piezo actuator and the cantilever along with the related mechanical mounting; and the dynamics uncertainties caused by the probe variation and the operation condition. In the proposed approach, an input force signal with frequency characteristics of band-limited white-noise is utilized to rapidly excite the nanomechanical response of materials over a broad frequency range. The MIIC technique is used to compensate for the hardware adverse effects, thereby allowing the precise application of such an excitation force and measurement of the material response (to the applied force). The proposed approach is illustrated by implementing it to measure the frequency-dependent plane-strain modulus of poly(dimethylsiloxane) (PDMS) over a broad frequency range extending over 3 orders of magnitude (~1 Hz to 4.5 kHz).     

Fer’s expansion for generalized Hamiltonian system based on Lie transformation technique

In the present paper, we present a new method for integrating the ordinary differential equation, especially for the ordinary differential equation derived from explicitly time-dependent generalized Hamiltonian dynamic system, which is based on taking a factorization of the evolution operator as an infinite product of the exponentials of Lie operators. The above process is a Lie group (algebraic) method that retains the structural intrinsic properties of the exact solution when truncated and is used to analyze the main features of the so-called Fer’s expansion. The numerical examples are presented at the end of this paper.     

Electro-mechanical coupling wave propagating in a locally resonant piezoelectric/elastic phononic crystal nanobeam with surface effects

The model of a "spring-mass" resonator periodically attached to a piezoelectric/elastic phononic crystal(PC) nanobeam with surface effects is proposed, and the corresponding calculation method of the band structures is formulized and displayed by introducing the Euler beam theory and the surface piezoelectricity theory to the plane wave expansion(PWE) method. In order to reveal the unique wave propagation characteristics of such a model, the band structures of locally resonant(LR) elastic PC Euler nanobeams with and without resonators, the band structures of LR piezoelectric PC Euler nanobeams with and without resonators, as well as the band structures of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on PZT-4, with resonators attached on epoxy, and without resonators are compared. The results demonstrate that adding resonators indeed plays an active role in opening and widening band gaps. Moreover, the influence rules of different parameters on the band gaps of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on epoxy are discussed, which will play an active role in the further realization of active control of wave propagations.