Nonlinear parametrically excited vibration and active control of gear pair system with time-varying characteristic

In the present work, we investigate the nonlinear parametrically excited vibration and active control of a gear pair system involving backlash, time-varying meshing stiffness and static transmission error. Firstly, a gear pair model is established in a strongly nonlinear form, and its nonlinear vibration characteristics are systematically investigated through different approaches. Several complicated phenomena such as period doubling bifurcation, anti period doubling bifurcation and chaos can be observed under the internal parametric excitation. Then, an active compensation controller is designed to suppress the vibration, including the chaos. Finally, the effectiveness of the proposed controller is verified numerically.

Vibration energy harvesting for low frequency using auto-tuning parametric rolling pendulum under exogenous multi-frequency excitations

An electromagnetic parametrically excited rolling pendulum energy harvester with self-tuning mechanisms subject to multi-frequency excitation is proposed and investigated in this paper.The system consists of two uncoupled rolling pendulum.The resonance frequency of each the rolling pendulum can be automatically tuned by adjusting its geometric parameters to access parametric resonance.This harvester can be used to harvest the energy at low frequency.A prototype is developed and evaluated.Its mathematical model is derived.A cam with rolling follower mechanism is employed to generate multi-frequency excitation.An experimental study is conducted to validate the proposed concept.The experimental results are confirmed by the numerical results.The harvester is successfully tuned when the angular velocity of the cam is changed from 1.149 to 1.236 Hz.

Parametrically excited oscillation of stay cable and its control in cable-stayed bridges＂

This paper presents a nonlinear dynamic model for simulation and analysis of a kind of parametrically excited vibration of stay cable caused by support motion in cable-stayed bridges. The sag, inclination angle of the stay cable are considered in the model, based on which, the oscillation mechanism and dynamic response characteristics of this kind of vibration are analyzed through numerical calculation. It is noted that parametrically excited oscillation of a stay cable with certain sag, inclination angle and initial static tension force may occur in cable-stayed bridges due to deck vibration under the condition that the natural frequency of a cable approaches to about half of the first model frequency of the bridge deck system. A new vibration control system installed on the cable anchorage is proposed as a possible damping system to suppress the cable parametric oscillation. The numerical calculation results showed that with the use of this damping system, the cable oscillation due to the vibration of the deck and/or towers will be considerably reduced.

Parametric Vibration of Submerged Floating Tunnel Tether Under Random Excitation

For the study of the parametric vibration response of submerged floating tunnel tether under random excitation, a nonlinear random parametric vibration equation of coupled tether and tube of submerged floating tunnel is set up. Subsequently, vibration response of tether in the tether-tube system is analyzed by Monte Carlo method. It may be concluded that when the tube is subjected to zero-mean Gaussian white noise random excitation, the displacement and velocity root mean square responses of tether reach the peak if the circular frequency of tube doubles that of tether; the displacement and velocity root mean square responses of tether increase as the random excitation root mean square increases; owing to the damping force of water, the displacement and velocity root mean square responses of tether decrease rapidly compared with tether in air; increasing the damping of the tether or tube reduces the displacement and velocity root mean square responses of tether; the large-amplitude vibration of tether may be avoided by locating dampers on the tether or tube.

Instability Assessment of Deep-Sea Risers Under Parametric Excitation

This study deals with the nonlinear dynamic response of deep-sea risers subjected to parametric excitation at the top of a platform. As offshore oil and gas exploration is pushed into deep waters, difficulties encountered in deep-sea riser design may be attributed to the existence of parametric instability regarding platform heave motions. Parametric resonance in risers can cause serious damage which might bring disastrous accidents such as environment pollution, property losses and even fatalities. Therefore, the paranletric instability analysis should attract more attention during the design process of deep-sea risers. In this work, an equation of motion for a deep-sea riser is derived firstly. The motion equation is analyzed by the Floquet theory which allows the determination of both system response and stability properties. The unstable regions in which parametric resonance easily occurs can be determined. The effects of damping on parametric instability are also investigated, and the stability maps are presented. The results demonstrate that the available damping is vital in suppressing the instability regions. The suggestions for reduction of instability regions are proposed in deep-sea riser design.

Stability and nonlinear vibrations of a flexible pipe parametrically excited by an internal varying flow density

Pipes are often used to transport multiphase flows in many engineering applications.The total fluid flow density inside a pipe may vary with time and space.In this paper,a simply supported pipe conveying a variable density flow is modeled theoretically,and its stability and nonlinear vibrations are investigated in detail.The variation of the flow density is simulated using a mathematical function.The equation governing the vibration of the pipe is derived according to Euler-Bernoulli beam theory.When the internal flow density varies with time,the pipe is excited parametrically.The stability of the pipe is determined by Floquet theory.Some simple parametric and combination resonances are determined.For a higher mass ratio(mean flow mass/pipe structural mass),higher flow velocity,or smaller end axial tension,the pipe becomes unstable more easily due to wider parametric resonance regions.In the subcritical flow velocity regime,the vibrations of the pipe are periodic and quasiperiodic for simple and combination resonances,respectively.However,in the supercritical regime,the vibrations of the pipe exhibit much richer dynamics including periodic,multiperiodic,quasiperiodic,and chaotic behaviors.

Parametric Vibration and Vibration Reduction of Cables in Cable-stayed Space Latticed Structure

Mechanical model and vibration equation of a cable in cable-stayed space latticed structure (CSLS) under external axial excitation were founded.Determination of the mass lumps and natural frequencies sup- plied by the space latticed structure (SLS) was analyzed.Multiple scales method (MSM) was introduced to analyze the characteristics of cable's parametric vibration,and the precise time-integration method (PTIM) was used to solve vibration equation.The vibration behavior of a cable is closely relative to the frequency ratio of the cable and SLS.The cable's parametric vibration caused by the external axial excitation easily occurs if the frequency ratio of the cable and SLS is in a certain range,and the cable's vibration amplitude varies greatly even if the initial disturbance supplied by SLS changes a little.Furthermore,the mechanical model and vibration equation of the composite cable system consisting of main cables and assistant cables were studied. The parametric analysis such as the pre-tension level and arrangement of the assistant cables was carried out. Due to the assistant cables,the single-cable vibration mode can be transferred to the global vibration mode, and the stiffness and damping of the cable system are enhanced.The natural frequencies of the composite cable system with the curve line arrangement of assistant cables are higher than those with the straight-line arrangement and the former is more effective than the latter on the cable's vibration suppression.

随机激励下斜拉索-梁参数振动模型及响应分析

为了揭示随机激励下斜拉索参数振动特性,考虑拉索几何非线性、倾角以及桥面梁的协同振动影响,建立了斜拉索-桥面梁耦合参数振动模型,利用相空间变换推导了索-梁结构耦合振动伊藤状态方程组,给出了高斯白噪声作用下索梁振动时程求解迭代格式,分析了拉索随机响应时程、频域、统计特性以及概率密度变化,并与高斯截断方法进行比较,考察了拉索阻尼比、初始张力、桥面梁的初始扰动以及随机激励强度对拉索振动的影响规律。结果表明:斜拉索振动迭代方法可有效求解随机激励下索-梁耦合参数振动时程,与传统的高斯截断方法的计算结果吻合;随机激励下,斜拉索“拍振”振幅具有明显的不确定性,斜拉索与桥面梁结构发生耦合振动时,两者之间存在能量传递现象;斜拉索随机激励位移响应均值具有非平稳瞬态特性,其位移均方差的稳定时间大于理想激励下的响应;斜拉索响应功率峰值对应的频率和功率谱密度旁瓣峰值衰减量基本一致,但斜拉索第一旁瓣峰值与主峰差值并不相同。斜拉索振动响应概率密度分布满足高斯分布形态,置信区间约为0.98,符合马尔科夫性质。随着阻尼比增大,拉索阻尼抑制效率降低;桥面梁初始扰动越大,拉索振动位移越大;拉索张力越大,拉索振动幅值呈非线性减小趋势,但张力进一步增大时,拉索振动幅值衰减趋于稳定;随机激励强度越大,拉索响应越大。

悬浮隧道管体运动对锚索涡激振动的影响规律

根据锚索式悬浮隧道的运动特性,假设锚索是两端铰接的非线性梁模型,并将管体运动简化为两自由度的质点运动,将其视作一种参数激励。考虑锚索长期处于海流环境下,其顺流向涡激力和管体横荡运动方向一致,锚索的运动响应可能会产生明显变化。建立锚索的非线性顺流向涡激振动方程,并通过Galerkin方法和Runge-Kutta方法对该方程进行求解,选取典型悬浮隧道的结构进行数值分析,结果表明:在锚索前三阶模态中,相比于一阶模态幅值,二阶和三阶模态幅值是一个很小的量,可以忽略不计;参数激励可以使锚索更快进入稳定状态,且对振动幅值的增加有显著影响;参数激励对顺流向涡激振动的影响程度与锚索顶端的端部激励幅值和频率有关。

旋转周期结构参激振动频率分裂研究

旋转周期结构广泛应用于机械工程领域,参激振动是其常见的振动形式。该文利用周期结构的时空对称特征提出了一种偏微分形式的时变弹性动力学模型,然后利用Galerkin方法及模态正交性得到了常微分形式的多自由度参激振动模型。研究了时变刚度激励作用下旋转周期结构的振动行为。为了研究参激振动的固频率分裂规律,采用调制反馈原理分析了不同反馈类型下的振动响应,揭示了时变刚度个数及振动波数等基本参数组合与频率分裂之间的映射关系,还预测了不同反馈类型下参激不稳定对应的激励频率,最后利用Floquét理论和Runge-Kutta方法分别验证了参数不稳定域及不同反馈类型下响应频率的正确性。

基于超细长弹性杆模型的斜拉索参数振动分析

将超细长弹性杆力学模型应用于描述大跨度斜拉桥拉索的非线性振动行为.首先,在假设拉索静态位形为悬链线型的基础上,考虑斜拉索的几何非线性、垂度和抗弯刚度,推导了拉索在轴向激励作用下的多模态非线性振动方程.其次,通过多尺度法对该振动方程求解,得到拉索发生主共振、主参数共振和3∶1次共振的常解存在条件;再根据Lyapunov近似稳定性判据进一步揭示振幅稳态解的渐进稳定条件.最后,通过近似解与数值解的对比分析,研究频率比、激励幅值和拉索阻尼对拉索振动特性的影响关系.结果表明:基于弹性杆模型的拉索振幅值略大于弹性梁模型,且拉索发生主参数共振的最小起振幅值有所降低;增大阻尼可一定程度上减小拉索的振动,但抑制效果有限,故控制激励幅值是减小拉索振动的有效方法.

FORCED SOLITARY WAVES IN A PARAMETRICALLY EXCITED CIRCULAR TANK

A new mode of nonpropagating solitary waves in circular tank, which are forcedly ex-cited in tbe vertical direction, is described. The waveform is measured and the data accordwell with the hyperbolic secant function.Analyses of the vibration modes of the wave showthat the mode is not a pure mode of circular container. A wax cone was inserted in themiddle of tank in order to keep down the plateau. The hysteresis phenomenon of excitation ofsolitary wave has been observed as well.

Modelling and Vibration of Gear Drive Systems Influenced by Real Face Width

Gear drives are one of the most common parts in many rotating machinery. If the gear drive runs under lower torque load, nonlinear effects like gear mesh interruption can occur and vibration is accompanied by impact motions of the gears, This paper presents an original method of the mathematical modelling of gear drive nonlinear vibrations by using the modal synthesis method with degrees of freedom number reduction. The model respects nonlinearities caused by gear mesh interruption, parametric gearing excitation caused by time-varying meshing stiffness and nonlinear contact forces acting between journals of the rolling-element bearings and the outer housing. The nonlinear model is then used for investigation of gear drive vibration, especially for constant gear mesh determination. The theoretical method is applied for investigating of test gear drive nonlinear vibration.

参外联合激励下非线性Zener系统的减振机理研究

旨在揭示参外联合激励下不同尺度非线性Zener系统的减振机理.以Duffing系统为主系统,引入周期变化的低频参数激励和外激励,通过耦合黏弹性元件,系统变为1.5自由度非线性Zener系统,经过对比系统变化前后时间历程图、相图,发现耦合黏弹性元件后,系统由单一激发态的大幅高频振动转变为激发态和沉寂态共存的簇发振动,且振动幅值大幅降低,减振效果明显.然后分析自治系统的稳定性和分岔情况,利用包络快慢分析法,将参数激励项定义为慢变参数,基于外激励在激励幅值变化范围内存在最值思想,分析了广义自治系统的稳定性、破缺分岔与非自治系统振动行为的密切关系.结果发现,自治系统对非自治系统具有明显的调节作用,具体表现为耦合黏弹性元件后自治系统平衡点稳定性增强,平衡点类型由中心变为稳定焦点,平衡线对系统轨线的吸引力增强,同时多条稳定平衡线限制了非自治系统的振动区域,这些因素是减振的根本原因.另外,基于双参数分岔分析,发现通过调节参数可以控制系统破缺分岔的发生,进而提高系统减振性能.

端部激励相位差对悬索亚谐波共振响应的影响

悬索受多个端部激励后可能产生复杂的非线性动力学现象.为研究端部激励相位差对悬索亚谐波响应的影响,建立两端受激励的悬索模型,得到无限维离散运动方程,采用多尺度法研究面内亚谐波共振,并通过数值积分加以验证.文中给出了多激励系统等效激励幅值和响应相位的一般表达式,定性定量分析了面内两两激励的不同组合对响应幅值和相位的影响.结果表明:端部激励相位差对悬索响应幅值的影响以2π为周期变化,关于π对称.端部激励间的相位差可以使响应相位在一个名义单激励系统的响应相位基础上发生移动,当激励相位差逐渐由0转变为π的同时,相频曲线也非同步地位移了π.激励相位差不改变系统的软硬性质,但会影响响应幅值和共振区宽度,且其共振区宽度关于σ=0对称.

张紧器结构非线性对深海顶张式立管系统参激稳定性的影响分析

顶张式立管在作业时,浮式平台垂荡运动导致立管轴向张力随时间变化,可能引发立管在水平方向发生参激失稳。随着海洋开发进入深海区,海洋环境愈加恶劣,立管因长径比大幅增加而柔性增强。为确保系统作业安全,必须寻找新的(以前被忽略或简化)关键影响因素以实现对立管动态响应准确预测和系统稳定性科学预判。在参激稳定性预报中,直接作用式张紧器最常采用的是线性弹簧模型和工程简化模型。考虑液压缸装置的气体压强和容积、摩擦力、液压损失等因素,建立了直接作用式张紧器非线性模型。采用张紧器线性弹簧模型、工程简化模型和非线性模型,基于富洛开理论预报1 000 m顶张式立管系统的参激稳定性。研究表明,张紧器结构非线性影响立管张紧力和固有频率计算,进而可能导致系统参激稳定误判。采用3种张紧器模型预报参激不稳定区的差别主要发生在低频区,利用非线性模型预报的参激不稳定区最大。随着蓄能器高压气体初始体积的增加和低压氮气瓶初始压强的提高,张紧器结构非线性效应会进一步加强。研究成果可用于修正现用的参激分析软件以更适用于深海区。

Nonlinear Dynamics of Torsional Vibration for Rolling Mill′s Main Drive System Under Parametric Excitation

The jointed shaft in the drivelines of the rolling mill, with its angle continuously varying in the production, has obvious impact on the stability of the main drive system. Considering the effect caused by the joint angle and friction force of roller gap, the nonlinear vibration model of the main drive system which contains parametric excitation stiffness and nonlinear friction damping was established. The amplitude-frequency characteristic equation and bifurcation response equation were obtained by using the method of multiple scales. Depending on the bifurcation response equation, the transition set and the topology structure of bifurcation curve of the system were obtained by using the singularity theory. The transition set can separate the system into seven areas, which has different bifurcation forms respectively. By taking the 1 780 rolling mill of Chengde Steel Co for example, the simulation and analysis were performed. The amplitude-frequency curves under different joint angles, damping coefficients, and nonlinear stiffness were given. The variations of these parameters have strong influences on the stability of electromechanical resonances and the characteristic of the response curves. The best angle of the jointed shaft is 4.761 3° in this rolling mill.

A novel adaptive harmonic balance method with an asymptotic harmonic selection

The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.

面内端部激励下斜拉索振动方程推导与求解

为精确描述面内端部激励下拉索非线性振动机制,构建抽象端部激励下的拉索力学描述模型,分别推导了拉索拟静态振动和模态振动控制方程,基于有限差分法给出了拉索模态振动的数值求解方法。由于垂度效应的影响,在轴向、竖向以及轴向和竖向共同激励下拉索的拟静态索力与端部位移采用一个非线性方程予以表征。拉索拟静态振动可视作外部激励施加在拉索上,引起拉索的非线性模态振动,通过理论推导分别求得模态振动的控制方程和变形协调方程。在模态振动过程中,拉索的动力特性(如刚度和周期)不断变化,索力和拉索变形相互耦合,使得其求解非常复杂。采用有限差分法进行数值求解,并基于有限元方法验证了数值求解方法的准确性。重点讨论了拉索模态振动共振机制,结果表明,在轴向、竖向以及轴向和竖向激励下,拉索模态振动存在多个共振区域,包括小周期、0.5 T 1、T 1和2T 1等(T 1为成桥状态下拉索基本振动周期);在共振区域内,拉索振动索力幅度较大,模态振动对总索力的贡献系数较高,超过0.4,在实际工程中不容忽视;拉索阻尼对模态振动的影响较大,增加拉索阻尼比将改变拉索模态振动的幅频特征,模态振动索力及其贡献系数均有所降低,但降低程度受激励幅度的影响较大。

中厚板轧机非线性参激的振动

建立了中厚板轧机两自由度非线性参激振动模型,分析了轧机振动系统在主参数共振情形下的振动特性,尤其是振动系统随激励参数变化的拟周期振动和混沌运动现象。分析结果表明:轧制过程中轧制力的周期性变化可以导致轧机振动系统混沌运动的产生;避免激振力频率与系统固有频率相近,以防止混沌的出现。