Simulation of the nonlinear vibration of a string using the Cellular Automata based on the reflection rule

In this study, the nonlinear dynamic responses of a string are simulated using the Cellular Automata method based on the reflection rule. In the case of nonlinear systems, the velocity of wave propagation is not constant and depends on the amplitude. A new treatment of the dynamic time step is proposed for the Cellular Automata method considering the effect of the propagation velocity. As numerical examples, first, the dynamic responses of a string with linear characteristic are simulated using the Cellular Automata method. A typical resonance curve can be obtained. Second, the dynamic responses of a string with nonlinear characteristic are simulated using the proposed method. Some characteristic types of vibration can be obtained. It is concluded that the linear and nonlinear dynamic responses of a string may be obtained by simulation using the Cellular Automata method.     

Analysis of energy dissipation in an elastic moving string with a viscous damper at one end

In this paper transverse vibration of an axially moving viscoelastic string with a viscous damper at one end is investigated analytically. The string is assumed to be travelling with constant velocity and the length of string is constant or time varying. The linear and nonlinear mathematical models are derived using the Lagrangian function and implemented using a finite element method. The method considers a time varying state space function applied to the linear model, the Newmark-Beta method is used to solve the response for the nonlinear problem numerically. The case of energy dissipated by a viscoelastic damper at one end of the string for different axial string velocities is considered. When a disturbance arrives at the boundary an exact value for the damper which provides maximum energy dissipation is investigated. Finally, numerical simulations are presented to establish the feasibility of the method.     

Vibration of strings with nonlinear supports

The dynamic string motion, which displacement is unilaterally constrained by the rigid termination condition of an arbitrary geometry has been simulated and analyzed. The treble strings of a grand piano usually terminate at a capo bar, which is situated above the strings. The apex of a V-shaped section of the capo bar defines the end of the speaking length of the strings. A numerical calculation based on the traveling wave solution is proposed for modeling the nonlinearity inducing interactions between the vibrating string and the contact condition at the point of string termination. It was shown that the lossless string vibrates in two distinct vibration regimes. In the beginning the string starts to interact in a nonlinear fashion with the rigid terminator, and the resulting string motion is aperiodic. Consequently, the spectrum of the string motion depends on the amplitude of string vibrations, and its spectral structure changes continuously with the passage of time. The duration of that vibration regime depends on the geometry of the terminator. After some time of aperiodic vibration, the string vibrations settle in a periodic regime where the resulting spectrum remains constant.     

Nonlinear amplitude approximation for bilinear systems

An efficient method to predict vibration amplitudes at the resonant frequencies of dynamical systems with piecewise-linear nonlinearity is developed. This technique is referred to as bilinear amplitude approximation (BAA). BAA constructs a single vibration cycle at each resonant frequency to approximate the periodic steady-state response of the system. It is postulated that the steady-state response is piece-wise linear and can be approximated by analyzing the response over two time intervals during which the system behaves linearly. Overall the dynamics is nonlinear, but the system is in a distinct linear state during each of the two time intervals. Thus, the approximated vibration cycle is constructed using linear analyses. The equation of motion for analyzing the vibration of each state is projected along the overlapping space spanned by the linear mode shapes active in each of the states. This overlapping space is where the vibratory energy is transferred from one state to the other when the system switches from one state to the other. The overlapping space can be obtained using singular value decomposition. The space where the energy is transferred is used together with transition conditions of displacement and velocity compatibility to construct a single vibration cycle and to compute the amplitude of the dynamics. Since the BAA method does not require numerical integration of nonlinear models, computational costs are very low. In this paper, the BAA method is first applied to a single-degree-of-freedom system. Then, a three-degree-of-freedom system is introduced to demonstrate a more general application of BAA. Finally, the BAA method is applied to a full bladed disk with a crack. Results comparing numerical solutions from full-order nonlinear analysis and results obtained using BAA are presented for all systems.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

圆孔非线性声阻抗三维时域仿真与特性分析

为研究有限振幅声波作用下圆孔的非线性声学特性,提出了基于三维时域计算流体动力学(CFD)仿真的圆孔非线性声阻抗提取方法,通过求解层流方程来模拟声信号在圆孔及上下游的传播,以及采用横向周期性边界条件来考虑高穿孔率时圆孔之间相互作用的影响。研究了不同幅值声波作用下孔径、厚度和穿孔率对声阻抗的影响规律,通过对质点振速幅值、频率和板厚等组成的无量纲参量进行非线性回归分析,得到了圆孔非线性声阻抗的拟合公式,并将其转换为可考虑多频声波影响的时域模型。最后结合声阻抗时域模型和有限差分方法计算了直通穿孔管消声器在小振幅和有限振幅声波作用下的传递损失,通过与实验测量结果的比较,验证了拟合公式的准确性和实用性。     

具有振幅调制的光学格子中光束的演化研究

利用变分原理和数值模拟的方法研究了光束在光传输方向的横截面上存在着具有振幅调制的光学格子时的演化,得到了光束宽度、振幅、光束的波阵面曲率随传播距离的演化形式.通过将光束的传输类比于粒子的运动,发现光束参量的条件存在三个区域,在不同的区域光束将在光格子的调制下有不同的传输形式,光格子振幅调制系数在各个区域起到了重要作用.光学格子中周期格子的作用有类似非线性的很好的特性,周期格子的引入提供了更好地控制格子孤子形成和传输的参考条件.     

次声波在非均匀运动大气中非线性传播特性的研究

基于有限时域差分方法将大气中近似到二阶的非线性波动方程进行离散化,得到了数值模拟所采用的差分方程. 在此基础上,对线阵列辐射的脉冲声波在非均匀运动大气中的垂直和斜向传播进行了二维数值模拟,模拟了武汉地区(114:20°E, 30:37°N)在夏季和冬季UT=29000 s时开始传播的脉冲声波在不同时刻的声压分布. 模拟时通过采用Msise00和HWM93 两个大气模型,考虑了由于大气温度和密度变化以及大气风场存在所引起的大气不均匀性和运动性. 通过研究上述两季有风与无风条件下的声压差值p_r,可以发现:风场对次声波在传播中声压分布的影响较大;由于不同季节和不同传播距离上"有效声速"的不同,导致了两季p_r分布波形存在差异;风场对声波非线性传播的影响要远大于其对线性传播的影响. 关键词： 次声波传播 非均匀运动大气 有效声速     

An iterative method to solve a regularized model for strongly nonlinear long internal waves

We present a simple iterative scheme to solve numerically a regularized internal wave model describing the large amplitude motion of the interface between two layers of different densities. Compared with the original strongly nonlinear internal wave model of Miyata [10] and Choi and Camassa [2], the regularized model adopted here suppresses shear instability associated with a velocity jump across the interface, but the coupling between the upper and lower layers is more complicated so that an additional system of coupled linear equations must be solved at every time step after a set of nonlinear evolution equations are integrated in time. Therefore, an efficient numerical scheme is desirable. In our iterative scheme, the linear system is decoupled and simple linear operators with constant coefficients are required to be inverted. Through linear analysis, it is shown that the scheme converges fast with an optimum choice of iteration parameters. After demonstrating its effectiveness for a model problem, the iterative scheme is applied to solve the regularized internal wave model using a pseudo-spectral method for the propagation of a single internal solitary wave and the head-on collision between two solitary waves of different wave amplitudes.     

Extending the dynamic range of an energy harvester using nonlinear damping

This paper introduces the use of nonlinear damping for extending the dynamic range of vibration energy harvesters. A cubic nonlinear damper is initially considered and the average harvested power and the throw are obtained for different sinusoidal base excitation amplitudes and frequencies, both numerically and analytically. It is demonstrated that when excited at resonance, at an amplitude below its maximum operational limit, the harvested power using a nonlinear damper can be significantly larger than that of a linear energy harvester, therefore expanding its dynamic range. A potential approach to implementing cubic nonlinearity using a shunted electromagnetic device is also presented.     

Influence of static and dynamic internal actions on elastic nonlinear properties of a granulated unconsolidated medium

We have studied the influence of external static (pressure) and dynamic (caused by an elastic wave with a finite amplitude) actions on the linear and nonlinear elastic properties of a granulated unconsolidated medium, which was simulated by steel spheres with diameters of 2 and 4 mm. We have analyzed the equation of state for such a medium taking into account the presence of weakly and strongly deformed contacts between individual spheres. We have obtained expressions for the elasticity coefficient and second- and third-order nonlinear elastic parameters, and we have experimentally studied the influence of external static pressure on their values. We have measured the dependence of the velocity of elastic waves on external static pressure and the probing signal amplitude. In the studied medium, a number of structural phase transitions were observed, related to rearrangement of the packing of spheres, which were caused by both static and dynamic actions. The structural phase transition was accompanied by an anomalous change in the nonlinear elastic parameters of the medium and the velocity of elastic waves. We have analyzed the results based on the Hertz theory of contact interaction.     

Ultrasonic waves in crystals with charged dislocations

A system of equations for charged dislocations, where the quadratic nonlinear terms are taken into account, is derived using the variational principle. This system describes the propagation of ultrasonic (US) waves in crystals with charged dislocations. From the linearized system of equations a linear dispersion equation is derived. Formulas for the phase linear velocity of the wave and the absorption coefficient are obtained, which show essential influence of charged dislocations and electrical properties of media on the mentioned quantities. For a nonlinear US wave an equation for the amplitude of the first harmonic is derived and, as a consequence, expressions are obtained for the nonlinear velocity of the US wave, for the attenuation of the first harmonic's amplitude, and for phase variation.     

动态应力场作用下线性啁啾光栅光谱特性研究

目前柔性结构损伤检测需多个传感器,且需要探测器具有极高的采集频率。光纤光栅具有动态响应快、易实现分布式测量等特点,为解决上述问题提供了有效的途径,提出采用线性啁啾光栅(LCFBG)实现动态应力场探测。首先,用传输矩阵理论建立了LCFBG反射光谱应变传感模型,用衰减正弦函数模拟沿LCFBG分布的动态应力场。通过仿真实验,详细研究了LCFBG反射光谱对不同振幅、不同衰减系数与不同传播速度动态应力场的响应特性。实验结果表明,LCFBG反射光谱的反射率、波长变化与光谱形状均与动态应力场上述参数有关,但是LCFBG反射光谱对动态应力场不同参数的响应规律不同。在一定范围内,LCFBG反射光谱的最大反射率随动态应力场幅值与速度的增大而增大,最终趋于一稳定值,但其随阻尼系数增大而减小。最后,研制了以LCFBG为敏感元件的传感器,并构建了动态应力场实验平台,实验结果与仿真实验数据基本一致。提供了一种通过实时采集LCFBG全光谱信息探测动态应力场的新方法。     

NON-LINEAR FREE VIBRATION ANALYSIS OF A STRING UNDER BENDING MOMENT EFFECTS USING THE PERTURBATION METHOD

In this paper, the non-linear free vibration of a string with large amplitude is considered. The initial tension, lateral vibration amplitude, cross-section diameter and the modulus of elasticity of the string have main effects on its natural frequencies. Increasing the lateral vibration amplitude makes the assumption of constant initial tension invalid. Therefore, it is impossible to use the classical equation of transverse motion assuming a small amplitude. On the other hand, by increasing the string cross-sectional diameter, the bending moment effect will increase dramatically, and it will act as an impressive restoring moment. Considering the effects of the bending moments, the non-linear equation governing the large amplitude transverse vibration of a string is derived. The time-dependent portion of the governing equation has the form of the Duffing equation. Due to the complexity and non-linearity of the derived equation and the fact that there is no established exact solution method, the equation is solved using the perturbation method. The results of the analysis are shown in appropriate graphs, and the natural frequencies of the string due to the non-linear factors are compared with the natural frequencies of the linear vibration of a string without bending moment effects.     

Prediction condensed models adapted to the nonlinear structures in time domain

This paper deals with a method to study closely the stationary solution of nonlinear dynamic systems in time domain. This method is based on the exploitation of Karhunen-Loève decomposition with or without parametric modifications as well as on the characteristics of localized nonlinearities. With the application of this method at first on linear models initially condensed by Karhunen-Loève, the predictions of nonlinear responses can be obtained rapidly. This method is adapted to a condensed linear model used in the first optimization procedure of the nonlinear dynamic behaviour. This robust basis will be used as condensation basis of the modified model local per zone, which leads to a prediction of vibratory responses of complex structures modified and affected by localized nonlinearities.     

A derating method for therapeutic applications of high intensity focused ultrasound

Current methods of determining high intensity focused ultrasound (HIFU) fields in tissue rely on extrapolation of measurements in water assuming linear wave propagation both in water and in tissue. Neglecting nonlinear propagation effects in the derating process can result in significant errors. A new method based on scaling the source amplitude is introduced to estimate focal parameters of nonlinear HIFU fields in tissue. Focal values of acoustic field parameters in absorptive tissue are obtained from a numerical solution to a KZK-type equation and are compared to those simulated for propagation in water. Focal wave-forms, peak pressures, and intensities are calculated over a wide range of source outputs and linear focusing gains. Our modeling indicates, that for the high gain sources which are typically used in therapeutic medical applications, the focal field parameters derated with our method agree well with numerical simulation in tissue. The feasibility of the derating method is demonstrated experimentally in excised bovine liver tissue.     

Method of Green’s function of nonlinear vibration of corrugated shallow shells

Based on the dynamic equations of nonlinear large deflection of axisymmetric shallow shells of revolution, the nonlinear free vibration and forced vibration of a corrugated shallow shell under concentrated load acting at the center have been investigated. The nonlinear partial differential equations of shallow shell were reduced to the nonlinear integral-differential equations by using the method of Green’s function. To solve the integral-differential equations, the expansion method was used to obtain Green’s function. Then the integral-differential equations were reduced to the form with a degenerate core by expanding Green’s function as a series of characteristic function. Therefore, the integral-differential equations became nonlinear ordinary differential equations with regard to time. The amplitude-frequency relation, with respect to the natural frequency of the lowest order and the amplitude-frequency response under harmonic force, were obtained by considering single mode vibration. As a numerical example, nonlinear free and forced vibration phenomena of shallow spherical shells with sinusoidal corrugation were studied. The obtained solutions are available for reference to the design of corrugated shells.     

超声波振动辅助下填充填缝工艺的数值研究

本文基于数值方法研究了超声波振动辅助倒装芯片成型下填充填缝工艺。采用了一种基于紧耦合的流固耦合算法,对结构和流场之间的耦合运动进行了数值模拟,其中流场部分采用有限体积法对任意拉格朗日-欧拉方法的不可压缩N-S方程进行离散计算,而结构部分采用有限元法对拉格朗日坐标下的弹性动力学方程进行离散计算。在一个时间步内,结构与流场的计算区域的交界面上进行了多次的数值传递和插值,以保证满足耦合面边界条件。计算了超声波振动作为边界条件作用于结构场,而引起的流场中流体流速的变化规律,并通过实验验证了本方法的正确性。同时,监测了流场入口体积流量的变化规律,探究了超声波振幅、频率以及流体粘度等因素对流体流速的影响,为超声波振动辅助倒装芯片成型下填充工艺的应用提供了理论依据。     

基于环结构的新型分布式光纤振动传感系统

提出并验证了一种新颖的基于环形马赫-泽德干涉仪结构的全分布式光纤振动传感系统.采用直流光实时动态定位技术，通过在干涉仪光路中引入环结构，将直线型干涉仪转化为环型回路，使得一个马赫-泽德干涉仪中相向传输两路光波，相当于构成双马赫-泽德干涉仪.当振动信号作用于传感光纤时，相向传输的两路直流光同时产生相同的相位信号并沿不同的路径传输至光接收单元，采用数字信号处理技术分析接收信号即可实时获得振动的空间位置和频率、幅度等特性参数.实验中成功实现了监测距离为1.01km的分布式振动传感，单点振动的空间分辨率小于40m.另外，从理论上分析并模拟了系统对多点同时振动进行检测和定位的可行性.     

Nonlinear solitary waves in nonlocal elastic solids

The possibility of new weakly nonlinear solitary waves in nonlocal elastic media is demonstrated. The properties of these waves are determined by the characteristic features of wave dispersion in the linear approximation, and their velocity and amplitude cannot exceed certain limiting values. In the case of small amplitudes and velocities close to the velocity of sound, the profile of the waves under consideration coincides with the profile of the soliton described by the Korteweg-de Vries equation. When the amplitude and velocity of the aforementioned waves reach their limiting values, the wave profile sharpens. It is concluded that the propagation of such waves in rocks and soils is possible.