Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.     

Effects of anti-symmetric nonlinear viscous damping on the force transmissibility of multi-degree of freedom structures

In the present study, the concept of the output frequency response function is applied to theoretically investigate the force transmissibility of multi-degree of freedom (MDOF) structures with a nonlinear anti-symmetric viscous damping. The results reveal that an anti-symmetric nonlinear viscous damping can significantly reduce the transmissibility over all resonance regions for MDOF structures while it has almost no effect on the transmissibility over non-resonant and isolation regions. The results indicate that the vibration isolators with an anti-symmetric damping characteristic have great potential to overcome the dilemma in the design of linear viscously damped vibration isolators where an increase of the damping level reduces the force transmissibility over resonant region but increases the transmissibility over non-resonant regions.     

旋转机械系统多自由度非线性动力学数值分析

首先将转子系统的动力响应问题归结为2n维未知向量v的一阶非线性动力学方程dv／dt=Hv＋f（v,t）,并给出了求解这一方程的一次近似式法和三次多项式迫近法。在非稳态、非线性油膜力等作用下,以刚性Jeffcott转子与112个自由度的汽轮发电机组低压转子系统为例,用上述求解方法分析了它们的动力响应及非线性动力学特性;其问,还将计算结果与Runge—Kutta法、Newmark法的相应结果进行了比较,并深入讨论了数值稳定性问题。汽轮发电机组的算例表明对一些具有较复杂的非线性右端项,、同时规模又较大的问题,如果采用四阶Runge—Kutta法,才算几步就因数值骤然增大而失控;但若用同样步长的一次近似式,由于它是一种显式的无条件稳定算式,则计算过程迅速且结果合理可靠。     

Wave propagation analysis in 2D nonlinear hexagonal periodic networks based on second order gradient nonlinear constitutive models

We analyze the propagation of nonlinear waves in homogenized periodic nonlinear hexagonal networks, considering successively 1D and 2D situations. Wave analysis is performed on the basis of the construction of the effective strain energy density of periodic hexagonal lattices in the nonlinear regime. The obtained second order gradient nonlinear continuum has two propagation modes: an evanescent subsonic mode that disappears after a certain wavenumber and a supersonic mode characterized by an increase of the frequency with the wavenumber. For a weak nonlinearity, a supersonic mode occurs and the dispersion curves lie above the linear dispersion curve (v_p =v_p0). For a higher nonlinearity, the wave changes from a supersonic to an evanescent subsonic mode at s=0.7 and the dispersion curves drops below the linear case and vanish for certain values of the wavenumber. An important decrease in the frequency occurs for both subsonic and supersonic modes when the lattice becomes auxetic, and the longitudinal and shear modes become very close to each other. The influence of the lattice geometrical parameters of the lattice on the dispersion relations is analyzed.     

Shear wave propagation in periodic phononic/photonic piezoelectric medium

Coupled electro-elastic SH waves propagating oblique to the lamination of a one dimensional piezoelectric periodic structure are considered in the framework of the full system of Maxwell’s electrodynamic equations. The dispersion equation has been obtained and numerical analyses carried out for two kinds of composites both consisting of two different piezoelectric materials. The results demonstrate the significant effect of piezoelectricity on the widths of band gaps at acoustic frequencies and confirm that it does not affect the band gaps at optical frequencies.     

Topology design and optimization of nonlinear periodic materials

This paper explores optimal topologies yielding large band gap shifts in one- and two-dimensional nonlinear periodic materials. The presence of a nonlinearity in a periodic material system results in amplitude-dependent dispersion behavior, leading to novel wave-based devices such as tunable filters, resonators, and waveguides. The performance of these devices over a broad frequency range requires large, tunable band gaps, motivating the present study. Consideration of a one-dimensional bilayer system composed of alternating linear and nonlinear layers shows that optimal designs consist of thin, compliant nonlinear layers. This is at first surprising considering the source of the shift originates from only the nonlinear layer; however, thin layers lead to localized stresses that activate the nonlinear character of the system. This trend persists in two-dimensional materials where optimization studies are performed on plane-stress models discretized using bilinear Lagrange elements. A fast algorithm is introduced for computing the dispersion shifts, enabling efficient parametric analyses of two-dimensional inclusion systems. Analogous to the one-dimensional system, it is shown that thin ligaments of nonlinear material lead to large dispersion shifts and group velocity variations. Optimal topologies of the two-dimensional system are also explored using genetic algorithms aimed at producing large increases in complete band gap width and shift, or group velocity variation, without presupposing the topology. The optimal topologies that result resemble the two-dimensional inclusion systems, but with small corner features that tend to enhance the production of dispersion shift further. Finally, the study concludes with a discussion on Bloch wave modes and their important role in the production of amplitude-dependent dispersion behavior. The results of the study provide insight and guidance on selecting topologies and materials which can yield large amplitude-dependent band gap shifts and group velocity variations, thus enabling sensitive nonlinear devices.     

Fully nonlinear modeling of radiated waves generated by floating flared structures

The nonlinear radiated waves generated by a structure in forced motion, are simulated numerically based on the potential theory. A fully nonlinear numerical model is developed by using a higher-order boundary element method （HOBEM）. In this model, the instantaneous body position and the transient free surface are updated at each time step. A Lagrangian technique is employed as the time marching scheme on the free surface. The mesh regridding and interpolation methods are adopted to deal with the possible numerical instability. Several auxiliary functions are proposed to calculate the wave loads indirectly, instead of directly predicting the temporal derivative of the velocity potential. Numerical experiments are carried out to simulate the heave motions of a submerged sphere in infinite water depth, the heave and pitch motions of a truncated flared cylinder in finite depth. The results are verified against the published numerical results to ensure the effectiveness of the proposed model. Moreover, a series of higher harmonic waves and force components are obtained by the Fourier transformation to investigate the nonlinear effect of oscillation frequency. The difference among fully nonlinear, body-nonlinear and linear results is analyzed. It is found that the nonlinearity due to free surface and body surface has significant influences on the numerical results of the radiated waves and forces.     

A coupled hydrodynamic–structural model of the M4 wave energy converter

A method is developed for modelling wave energy converters consisting of floats connected by slender structural elements. The hydrodynamic and structural dynamic analyses are separated in a two-stage process, though the model is fully coupled. The method of dynamic substructuring is used to achieve this separation. The linear diffraction/radiation problem is solved with a finite element idealisation for axisymmetric floats, and drag forces are incorporated by equivalent linearization. Results for a planar representation of the M4 device, and comparisons of theory and experiments undertaken for two scale models tested in regular and random waves, confirm the validity of the theoretical approach. A series of parametric studies is performed to clarify the important physical variables, including natural periods, the ratio of a characteristic length of the device to the wave length, and power take-off.     

Energy method for computing periodic solutions of strongly nonlinear systems (I) — Autonomous systems

In this paper a new method for solving for the periodic solution (limit cycle) of a strongly nonlinear system is suggested. Using this method not only the existence, stability and number of periodic solutions can be decided, but at the same time the approximate expressions for these periodic solutions can also be obtained. The proof of this method is given as well.The project is supported by the National Natural Science Foundation of China.     

Transient wave-blockage interaction and extended blockage detection in elastic water pipelines

Extended partial blockages are common in pressurized water pipelines and can result in the wastage of energy, the reduction in system carrying capacity and the increased potential for contamination. This paper investigates the transient wave-blockage interaction and its application to extended blockage detection in pipelines, where blockage-induced changes to the system resonant frequencies are observed. The frequency shifting is first inspected and explained in this study through wave perturbation analysis, providing a theoretical confirmation for the result that unlike discrete blockages, extended blockages cause resonant frequency shifts in the system. Furthermore, an analytical expression is derived for the relationship between the blockage properties and the resonant frequency shifts and is used to detect the blockages in this study. The obtained results are validated through both numerical applications and laboratory experiments, where the accuracy and efficiency of the developed method for extended blockage detection are tested.     

Active control of elastic wave propagation in nonlinear phononic crystals consisting of diatomic lattice chain

This work studies the active control effects on nonlinear phononic crystals by the piezoelectric spring model. Both negative and positive proportional control actions are considered. Based on the Lindstedt–Poincaré method, the approximate solution is derived and the stop band properties are presented. Numerical calculations show that the structural stiffness and negative proportional control of the piezoelectric spring can create a new stop band which is under the acoustic branch. But the positive proportional case has a different influence because of a critical wave number appearing. Moreover, the optic branch can be uplifted by the elastic wave amplitude. Different from the soft nonlinear characteristic, the hard nonlinear property can increase the stop band width.     

On the analysis of 2D nonlinear gravity waves with a fully nonlinear numerical model

A fully nonlinear numerical method, developed on the basis of Euler equations, is used to study the dynamics of nonlinear gravity waves, mainly in the aspects of the propagation of Stokes wave with disturbed sidebands, the evolution of one wave packet and the interaction of two wave groups. These cases have previously been studied with the higher order spectral method, which will be an approximately fully nonlinear scheme if the order of nonlinearity is not large enough, while the present method in the case of the 2D model has an integration scheme that is exact to the computer precision. As expected, in most cases the results are consistent between these two numerical models and it is confirmed again that this fully nonlinear numerical model is also capable of maintaining a high accuracy and good convergence, particularly in the long-term evolutionary process.     

钢筋混凝土框架—剪力墙结构非线性抗震分析的一种空间力学模型

将钢筋混凝土框架－剪力墙结构离散为能模拟梁、柱、墙抗震性能的单元，采用杆系－层间模型进行结构的非线性抗震分析。本文的空间力学模型可考虑局部楼板变形对结构地震反应的影响，并可沿任意角度输入相互垂直的两个地震动水平分量，适用于框架一剪力墙复杂结构的非线性抗震分析。     

Novel numerical implementation of asymptotic homogenization method for periodic plate structures

The present paper develops and implements finite element formulation for the asymptotic homogenization theory for periodic composite plate and shell structures, earlier developed in  and , and thus adopts this analytical method for the analysis of periodic inhomogeneous plates and shells with more complicated periodic microstructures. It provides a benchmark test platform for evaluating various methods such as representative volume approaches to calculate effective properties. Furthermore, the new numerical implementation (Cheng et al., 2013) of asymptotic homogenization method of 2D and 3D materials with periodic microstructure is shown to be directly applicable to predict effective properties of periodic plates without any complicated mathematical derivation. The new numerical implementation is based on the rigorous mathematical foundation of the asymptotic homogenization method, and also simplicity similar to the representative volume method. It can be applied easily using commercial software as a black box. Different kinds of elements and modeling techniques available in commercial software can be used to discretize the unit cell. Several numerical examples are given to demonstrate the validity of the proposed methods.     

Existence of time periodic solutions for a damped generalized coupled nonlinear wave equations

IntroductionInRef.[1 ] ,theauthorsestablishedtheuniqueexistenceofthesmoothsolutionforthefollowingcouplednonlinearequationsut=uxxx+buux+ 2vvx, (1 )vt=2 (uv) x. (2 )Thesewereproposedtodescribetheinteractionprocessofinternallongwaves.InRef.[2 ] ,ItoM .proposedarecursionoperatorbywhichheinferredthatEqs.(1 )and (2 )possesinfinitelymanysymmetriesandconstantsofmotion .InRef.[3 ] ,P .F .HeestablishedtheexistenceofasmoothsolutiontothesystemofcouplednonlinearKdVequation[4 ]ut=a(uxxx+buux) + 2bvvx,(…     

Dispersion characteristics of wave propagation in layered piezoelectric structures: Exact and simplified models

This article presents a study of the dispersion characteristics of wave propagation in layered piezoelectric structures under plane strain and open-loop conditions. The exact dispersion relation is first determined based on an electro-elastodynamic analysis. The dispersion equation is complicated and can be solved only by numerical methods. Since the piezoelectric layer is very thin and can be modeled as an electro-elastic film, a simplified model of the piezoelectric layer reduces this complex problem to a non-trivial solution of a series of quadratic equations of wave numbers. The model is simple, yet captures the main phenomena of wave propagation. This model determines the dispersion curves of PZT4-Aluminum layered structures and identifies the two lowest modes of waves: the generalized longitudinal mode and the generalized Rayleigh mode. The model is validated by comparing with exact solutions, indicating that the results are accurate when the thickness of the layer is smaller or comparable to the typical wavelength. The effect of the piezoelectricity is examined, showing a significant influence on the generalized longitudinal wave but a very limited effect on the generalized Rayleigh wave. Typical examples are provided to illustrate the wave modes and the effects of layer thickness in the simplified model and the effects of the material combinations.     

利用叠加理论研究固-液声子晶体纵波的能带结构

采用相干叠加原理推导出一种研究一维固-液有限周期声子晶体中纵波的透射率公式,建立了一种研究一维固-液有限周期声子晶体能带的新方法——波叠加理论。将波叠加理论和转移矩阵法进行了比较研究,结果表明波叠加理论和转移矩阵法得出的结果是一致的。波叠加理论不仅具有转移矩阵法的优点,而且又克服了转移矩阵法的不足。因此波叠加理论是一种研究一维固-液有限周期声子晶体中纵波能带的更为有效的方法。     

Weakly nonlinear long waves in a prestretched Blatz-Ko cylinder: Solitary, kink and periodic waves

This paper studies nonlinear waves in a prestretched cylinder composed of a Blatz-Ko material. Starting from the three-dimensional field equations, two coupled PDEs for modeling weakly nonlinear long waves are derived by using the method of coupled series and asymptotic expansions. Comparing with some other existing models in literature, an important feature of these equations is that they are consistent with traction-free surface conditions asymptotically. Also, the material nonlinearity is kept to the third order. As these two PDEs are quite complicated, the attention is focused on traveling waves, for which a first-order system of ODEs are obtained. We use the technique of dynamical systems to carry out the analysis. It turns out that the system is three parameters (the prestretch, the propagating speed and an integration constant) dependent and there are totally seven types of phase planes which contain trajectories representing bounded traveling waves. The parametric conditions for each phase plane are established. A variety of solitary and periodic waves are found. An important finding is that kink waves can propagate in a Blatz-Ko cylinder. We also find that one type of periodic waves has an interesting feature in the profile slope. Analytical expressions for all bounded traveling waves are obtained.