Transient analysis of nonlinear Euler–Bernoulli micro-beam with thermoelastic damping,via nonlinear normal modes

In this paper an Euler–Bernoulli model has been used for vibration analysis of micro-beams with large transverse deflection. Thermoelastic damping is considered to be the dominant damping mechanism and introduced as imaginary stiffness into the equation of motion by evaluating temperature profile as a function of lateral displacement. The obtained equation of motion is analyzed in the case of pure single mode motion by two methods; nonlinear normal mode theory and the Galerkin procedure. In contrast with the Galerkin procedure, nonlinear normal mode analysis introduces a nonconventional nonlinear damping term in modal oscillator which results in strong damping in case of large amplitude vibrations. Evaluated modal oscillators are solved using harmonic balance method and tackling damping terms introduced as an imaginary stiffness is discussed. It has been shown also that nonlinear modal analysis of micro-beam with thermoelastic damping predicts parameters such as inverse quality factor, and frequency shift, to have an extrema point at certain amplitude during transient response due to the mentioned nonlinear damping term; and the effect of system?s characteristics on this critical amplitude has also been discussed.     

Factorizing coupled quantum systems and damped nonlinear Schrödinger equations

We show that the wave function of a coupled quantum system may factorize for certain coupling operators, resulting in wave functions and effective nonlinear Hamiltonians for the subsystems. Systems of coupled harmonic oscillators with discrete or continuous spectra are considered, where all degrees of freedom move in time-dependent coherent Glauber states.We present the general formalism and study two examples in detail. The problem of radiation damping results under drastic assumptions in exponentially damped harmonic motion, obeying a nonlinear Schrödinger equation. In the second example, a different type of coupling is studied which yields inverse power law damping.     

掩埋隧道结垂直面激光器速率方程参量提取

为了设计和优化高速激光二极管的高频特性,其速率方程模型参量的精确提取方法非常重要.本文针对新型长波长高带宽的掩埋隧道结垂直面激光器,给出一种速率方程模型参量提取方法.此方法是主要基于阈值电流、输出光功率、张弛振荡频率、阻尼因子和高偏置下增益压缩因子非线性效应等因素,根据已测量的不同偏置下芯片的小信号频率响应来拟合出方程中的张弛振荡频率和阻尼因子.通过考虑增益压缩因子,分别非线性拟合已提取的偏置光功率下的张弛振荡频率和阻尼因子,即可提取速率方程模型中的参量.     

A decrement method for quantifying nonlinear and linear damping parameters

A method is presented which can estimate the linear and nonlinear damping parameters in a lightly damped system. Only a single response measurement from a free decay test is required as input. This ensures that the magnitude of the damping parameters is not compromised by phase distortion between measurements. The method uses the instantaneous energy to describe the long-term evolution of the system. Practically this is achieved by using only the peak amplitudes in each period. In this way the stiffness is effectively ignored, and only the damping forces are considered. For this reason, the method is not unlike the familiar decrement method, which can be used to estimate the apparent linear damping. The method is developed in the context of a weakly nonlinear, lightly damped system, with both linear and cubic damping. Simulated response data is used to demonstrate the accuracy of the technique. The nonlinear damping parameter is extracted from the response data to within 5% of the exact value, even though the nonlinear term contributes less than 1% to the total force in the system.     

Plasma wave frequency shift in a weak transverse magnetic field due to trapped particle acceleration

A simple model of nonlinear electrostatic wave–particle interaction in a weak magnetic field perpendicular to the direction of wave propagation is developed. The damping of the wave loaded with the phase bunched groups of trapped particles is considered with the aid of the model equations. To determine the nonlinear frequency shift of the wave in the process of the trapped particle acceleration, the nonlinear dispersion equation is derived. It is shown that the corresponding variation of the phase velocity may affect the interaction process and hence must be taken into account in the self-consistent treatment of the time evolution of the wave.     

柴油机轴系扭振系统的强非线性主共振

研究柴油机轴系扭振系统强非线性问题.根据拉格朗日方程建立柴油机轴系扭振系统的动力学模型,通过参数变换,应用Modified Lindstedt-Poincaré方法得到柴油机轴系扭振系统强非线性主共振的幅频响应方程,分析系统不同参数对主共振幅频响应的影响.结果表明,系统的幅频响应曲线存在跳跃,随着简谐力矩的减小和阻尼的增大,系统的非线性跳跃减弱,系统的振幅减小,系统主共振的区域也随之减小;随着调谐参数的变化,系统的主共振力幅响应曲线存在两种拓扑结构.MLP方法得出的近似解析解与龙格库塔法得出的数值解吻合.     

Stability of a nonlinear oscillator with random damping

A noisy damping parameter in the equation of motion of a nonlinear oscillator renders the fixed point of the system unstable when the amplitude of the noise is sufficiently large. However, the stability diagram of the system can not be predicted from the analysis of the moments of the linearized equation. In the case of a white noise, an exact formula for the Lyapunov exponent of the system is derived. We then calculate the critical damping for which the nonlinear system becomes unstable. We also characterize the intermittent structure of the bifurcated state above threshold and address the effect of temporal correlations of the noise by considering an Ornstein-Uhlenbeck noise.     

Comments on nonlinear dynamics of a non-ideal Duffing-Rayleigh oscillator: Numerical and analytical approaches

An analytical and numerical investigation into the dynamic interaction between a cantilever beam with nonlinear damping and stiffness behavior, modeled by the Duffing-Rayleigh equation, and a non-ideal motor that is connected to the end of the beam, is presented. Non-stationary and steady-state responses in the resonance region as well as the passage through resonance behavior when the frequency of the excitation is varied are analyzed. The influences of nonlinear stiffness, nonlinear damping and the extent of the unbalance in the motor are examined. It is found that in this situation so-called Sommerfeld effects may be observed; the increase required by a source operating near the resonance results in a small change in the frequency, but there is a large increase in the amplitude of the resultant vibration and the jump phenomenon occurs.     

基于Coulomb摩擦效应的一类非线性相对转动系统的混沌研究

研究一类非线性相对转动系统在负载Coulomb摩擦效应下的混沌运动行为. 根据Lagrange方程建立一类含非线性负载Coulomb摩擦阻尼的两个质量相对转动系统的动力学方程. 利用Cardano公式讨论自治系统的特征值, 在此基础上, 应用待定系数法给出系统同宿轨道的存在性, 并借助Silnikov定理研究了系统的混沌行为. 最后数值模拟了给定参数下系统的混沌运动, 并给出在Coulomb摩擦阻尼变化下系统由周期、倍周期通向混沌的途径, 验证了理论分析的正确性.     

Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model

Linear models and synchronous response are generally adequate to describe and analyze rotors supported by hydrodynamic bearings. Hence, stiffness and damping coefficients can provide a good model for a wide range of situations. However, in some cases, this approach does not suffice to describe the dynamic behavior of the rotor-bearing system. Moreover, unstable motion occurs due to precessional orbits in the rotor-bearing system. This instability is called “oil whirl” or “oil whip”. The oil whirl phenomenon occurs when the journal bearings are lightly loaded and the shaft is whirling at a frequency close to one-half of rotor angular speed. When the angular speed of the rotor reaches approximately twice the natural frequency (first critical speed), the oil whip phenomenon occurs and remains even if the rotor angular speed increases. Its frequency and vibration mode correspond to the first critical speed. The main purpose of this paper is to validate a complete nonlinear solution to simulate the fluid-induced instability during run-up and run-down. A flexible rotor with a central disk under unbalanced excitation is modeled. A nonlinear hydrodynamic model is considered for short bearing and laminar flow. The effects of unbalance, journal-bearing parameters and rotor arrangement (vertical or horizontal) on the instability threshold are verified. The model simulations are compared with measurements at a real vertical power plant and a horizontal test rig.     

Strongly nonlinear shear perturbations in discrete and continuous cubic nonlinear systems

Systems with constraints, the masses in which move only along guides, can execute strongly nonlinear vibrations. This means that nonlinear phenomena manifest themselves at arbitrary small deviations from equilibrium. The form of vibrations of a single mass is described by elliptical Jacobi functions. The spectrum of these vibrations is found. With an increase in amplitude, the period of vibrations decreases. We deduce equations of strongly nonlinear vibrations of a chain of connected masses. In the continuum limit, we obtain a new nonlinear equation in partial derivatives. We devise transformation of variables leading to linearization of this equation. We implemented a factorization procedure that decreases the order of the equation in partial derivatives from second to first. Exact solutions to the first-order equation describe the slow evolution of the displacement profile in a distributed system. In the absence of preliminary tension of elastic elements in the continued model, traveling waves cannot be achieved; however, time-oscillating solutions like standing waves are possible. We obtain an equation for a field of strongly nonlinear deformations. Its exact solution describes periodic movement in time and space. As well, the period of time oscillations decreases with an increase in amplitude, and the spatial period, in contrast, increases. The product of the vibration frequency multiplied by the spatial period is a constant that depends on the deformation energy. We propose a scheme of the mechanical system producing strongly nonlinear torsional vibrations. We experimentally measured the period of torsional vibrations of a single disc. We show that with an increase in amplitude in the process of vibration attenuation, an increase in the period occurs, which agrees with calculations. We measure the shapes of nonlinear vibrations of a chain of connected discs. A strongly nonlinear behavior of the chain is observed.     

Post-buckling longterm dynamics of a forced nonlinear beam: A perturbation approach

The aim of this paper is to determine by a singular perturbation approach the dynamic response of a harmonically forced system experiencing a pitchfork bifurcation. The model of an extensible beam forced by a harmonic excitation and subject to an axial static buckling is space-discretized by a Galerkin approach and studied by the Normal Form Method for different values of equation parameters influencing the nonlinear dynamic behavior like damping coefficient, load amplitude and frequency. A relevant issue in the perturbation methods is the concept of small and zero divisors which are related to the possibility to build a transformation that simplifies the original studied problem, i.e. to obtain the Normal Form, by eliminating as much as possible nonlinearities in the equations. For nonconservative systems, like structural damped systems, there are no conditions in the prior literature that define what “small” means relatively to a divisor. In the present paper some conditions about the order of magnitude of the divisors with respect to the perturbation entity are given and related to some physical parameters in the governing equations in order to estimate the relevance of some nonlinear effects.     

M-shaped asymmetric nonlinear oscillator for broadband vibration energy harvesting: Harmonic balance analysis and experimental validation

Over the past few years, nonlinear oscillators have been given growing attention due to their ability to enhance the performance of energy harvesting devices by increasing the frequency bandwidth. Duffing oscillators are a type of nonlinear oscillator characterized by a symmetric hardening or softening cubic restoring force. In order to realize the cubic nonlinearity in a cantilever at reasonable excitation levels, often an external magnetic field or mechanical load is imposed, since the inherent geometric nonlinearity would otherwise require impractically high excitation levels to be pronounced. As an alternative to magnetoelastic structures and other complex forms of symmetric Duffing oscillators, an M-shaped nonlinear bent beam with clamped end conditions is presented and investigated for bandwidth enhancement under base excitation. The proposed M-shaped oscillator made of spring steel is very easy to fabricate as it does not require extra discrete components to assemble, and furthermore, its asymmetric nonlinear behavior can be pronounced yielding broadband behavior under low excitation levels. For a prototype configuration, linear and nonlinear system parameters extracted from experiments are used to develop a lumped-parameter mathematical model. Quadratic damping is included in the model to account for nonlinear dissipative effects. A multi-term harmonic balance solution is obtained to study the effects of higher harmonics and a constant term. A single-term closed-form frequency response equation is also extracted and compared with the multi-term harmonic balance solution. It is observed that the single-term solution overestimates the frequency of upper saddle-node bifurcation point and underestimates the response magnitude in the large response branch. Multi-term solutions can be as accurate as time-domain solutions, with the advantage of significantly reduced computation time. Overall, substantial bandwidth enhancement with increasing base excitation is validated experimentally, analytically, and numerically. As compared to the 3 dB bandwidth of the corresponding linear system with the same linear damping ratio, the M-shaped oscillator offers 3200, 5600, and 8900 percent bandwidth enhancement at the root-mean-square base excitation levels of 0.03g, 0.05g, and 0.07g, respectively. The M-shaped configuration can easily be exploited in piezoelectric and electromagnetic energy harvesting as well as their hybrid combinations due to the existence of both large strain and kinetic energy regions. A demonstrative case study is given for electromagnetic energy harvesting, revealing the importance of higher harmonics and the need for multi-term harmonic balance analysis for predicting the electrical power output accurately.     

Asymmetric wave propagation in nonlinear systems

A mechanism for asymmetric (nonreciprocal) wave transmission is presented. As a reference system, we consider a layered nonlinear, nonmirror-symmetric model described by the one-dimensional discrete nonlinear Schr?dinger equation with spatially varying coefficients embedded in an otherwise linear lattice. We construct a class of exact extended solutions such that waves with the same frequency and incident amplitude impinging from left and right directions have very different transmission coefficients. This effect arises already for the simplest case of two nonlinear layers and is associated with the shift of nonlinear resonances. Increasing the number of layers considerably increases the complexity of the family of solutions. Finally, numerical simulations of asymmetric wave packet transmission are presented which beautifully display the rectifying effect.     

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over "fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.     

一类相对转动时滞非线性动力系统的稳定性分析

建立了具有时变刚度、非线性阻尼和谐波激励的一类相对转动时滞非线性动力系统的动力学方程.采用多尺度法推导出时滞动力系统的分岔响应方程,运用奇异性理论研究系统结构稳定性,得到主共振稳态响应方程的转迁集以及不同参数下分岔曲线的拓扑结构.应用Hopf分岔理论讨论了时滞动力系统动态稳定性,给出了系统产生极限环的条件,最后用数值模拟的方法研究了时滞参数对系统极限环幅值的影响。     

Finite-amplitude standing acoustic waves in a cubically nonlinear medium

The acoustic field in a resonator filled with a cubically nonlinear medium is investigated. The field is represented as a linear superposition of two strongly distorted counterpropagating waves. Unlike the case of a quadratically nonlinear medium, the counterpropagating waves in a cubically nonlinear medium are coupled through their mean (over a period) intensities. Free and forced standing waves are considered. Profiles of discontinuous oscillations containing compression and expansion shock fronts are constructed. Resonance curves, which represent the dependences of the mean field intensity on the difference between the boundary oscillation frequency and the frequency of one of the resonator modes, are calculated. The structure of the profiles of strongly distorted “forced” waves is analyzed. It is shown that discontinuities are formed only when the difference between the mean intensity and the detuning takes certain negative values. The discontinuities correspond to the jumps between different solutions to a nonlinear integro-differential equation, which, in the case of small dissipation, degenerates into a third-degree algebraic equation with an undetermined coefficient. The dependence of the intensity of discontinuous standing waves on the frequency of oscillations of the resonator boundary is determined. A nonlinear saturation is revealed: at a very large amplitude of the resonator wall oscillations, the field intensity in the resonator ceases depending on the amplitude and cannot exceed a certain limiting value, which is determined by the nonlinear attenuation at the shock fronts. This intensity maximum is reached when the frequency smoothly increases above the linear resonance. A hysteresis arises, and a bistability takes place, as in the case of a concentrated system at a nonlinear resonance.     

一类相对转动非线性动力系统的混沌运动

研究一类具有同宿轨道、异宿轨道的相对转动非线性动力系统的混沌运动. 建立具有非线性刚度、非线性阻尼和外扰激励作用的一类两质量相对转动非线性动力系统的动力学方程. 利用Melnikov方法讨论了系统的全局分岔和系统进入混沌状态的可能途径,给出了系统发生混沌的必要条件,并利用最大Lyapunov指数图,分岔图,Poincare截面图和相轨迹图进一步分析了系统的混沌行为. 关键词： 相对转动 非线性动力系统 混沌 Melnikov方法     

A nonlinear POD reduced order model for limit cycle oscillation prediction

As the amplitude of the unsteady flow oscillation is large or large changes occur in the mean background flow such as limit cycle oscillation,the traditional proper orthogonal decomposition reduced order model based on linearized time or frequency domain small disturbance solvers can not capture the main nonlinear features.A new nonlinear reduced order model based on the dynamically nonlinear flow equation was investigated.The nonlinear second order snapshot equation in the time domain for proper orthogonal decomposition basis construction was obtained from the Taylor series expansion of the flow solver.The NLR 7301 airfoil configuration and Goland+ wing/store aeroelastic model were used to validate the capability and efficiency of the new nonlinear reduced order model.The simulation results indicate that the proposed new reduced order model can capture the limit cycle oscillation of aeroelastic system very well,while the traditional proper orthogonal decomposition reduced order model will lose effectiveness.