Novel method for random vibration analysis of single-degree-of-freedom vibroimpact systems with bilateral barriers

The vibroimpact systems with bilateral barriers are often encountered in practice.However,the dynamics of the vibroimpact system with bilateral barriers is full of challenges.Few closed-form solutions were obtained.In this paper,we propose a novel method for random vibration analysis of single-degree-of-freedom(SDOF)vibroim-pact systems with bilateral barriers under Gaussian white noise excitations.A periodic approximate transformation is employed to convert the equations of the motion to a con-tinuous form.The probabilistic description of the system is subsequently defined through the corresponding Fokker-Planck-Kolmogorov(FPK)equation.The closed-form station-ary probability density function(PDF)of the response is obtained by solving the reduced FPK equation and using the proposed iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow.Finally,the versatility of the proposed approach is demonstrated by its application to two typical examples.Note that the solution obtained by using the proposed method can be used as the benchmark to examine the accuracy of approximate solutions obtained by other methods.

Response of train-bridge system under intensive seismic excitation by random vibration method

Earthquake is a kind of sudden and destructive random excitation in nature.It is significant to determine the probability distribution characteristics of the corresponding dynamic indicators to ensure the safety and the stability of structures when the intensive seismic excitation,the intensity of which is larger than 7,acts in train-bridge system.Firstly,the motion equations of a two-dimensional train-bridge system under the vertical random excitation of track irregularity and the vertical seismic acceleration are established,where the train subsystem is composed of 8 mutually independent vehicle elements with 48 degrees of freedom,while the single-span simple supported bridge subsystem is composed of 102D beam elements with 20 degrees of freedom on beam and 2 large mass degrees of freedom at the support.Secondly,Monte Carlo method and pseudo excitation method are adopted to analyze the statistical parameters of the system.The power spectrum density of random excitation is used to define a series of non-stationary pseudo excitation in pseudo excitation method and the trigonometric series of random vibration history samples in Monte Carlo method,respectively solved by precise integral method and Newmark-βmethod through the inter-system iterative procedure.Finally,the results are compared with the case under the weak seismic excitation,and show that the samples of vertical acceleration response of bridge and the offload factor of train obeys the normal distribution.In a high probability,the intensive earthquakes pose a greater threat to the safety and stability of bridges and trains than the weak ones.

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitudefrequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.

AN EQUIVALENT NONLINEARIZATION METHOD FOR ANALYSINGRESPONSE OF NONLINEAR SYSTEMS TO RANDOM EXCITATIONS

In this paper, a new equivalent nonlinearization method is developed and used in analysing the response of nonlinear systems to Gaussian while noise excitation. Its basic idea and calculation method are expounded. With the help of the presented method, several kinds of usual nonlinear random vibration systems are analyzed. The numerical results show that the mean square responses of the proposed approach are much closer to the exact solutions or Monte Carlo solutions, than that obtained from equivalent linearization method.

弱非线性系统下的耗能减震结构随机地震响应分析

本文研究了结构刚度弱非线性和滞变弱非线性的耗能减震结构的随机地震响应问题。首先利用整体本构关系建立单自由度弱非线性耗能结构的运动方程;然后结合带支撑黏弹性阻尼器的等效阻尼原理和随机等效线性化法对其进行等效;最后基于FPK(Fokker-Planck-Kolmogorov)方程法推导出结构刚度弱非线性耗能结构随机地震响应值的解析解,基于等效Duffing体系法推导出结构刚度滞变弱非线性耗能结构随机地震响应值的解析解,并对这2种方法的误差进行了分析。算例分析表明:计算结果可靠、有效;相较于现有的随机等效线性化法的求解方法,FPK法和等效Duffing体系法具有更高的精度。

协方差分析描述函数法的通用化数值算法

协方差分析描述函数法(covariance analysis describing function technique,CADET)在处理系统的随机响应问题上具有求解迅速、仿真精度高等优点.但对于复杂系统,其理论推导过程、求解系统解析响应方程较为复杂繁琐.为进一步推广CADET的应用,依托高斯–埃尔米特积分法,提出了一种通用化的CADET数值算法.作为算法验证,以车辆行驶过程中的随机振动为例,建立了几种不同非线性悬架车辆的二自由度动力学模型,并将CADET通用化数值算法与传统CADET算法及蒙特卡罗法进行了对比分析.仿真结果表明,CADET的通用化数值算法可以达到满足应用要求的计算精度,这验证了所提数值算法的有效性,且具有更强的泛化应用于复杂非线性动力系统的价值.

Fokker-Planck方程的算子分裂算法

为减少传统Runge-Kutta有限单元法求解福克-普朗克(FP)方程所需的冗长时间,提出一种有限单元法结合算子分裂法的FP方程数值求解新方法。该方法通过对有限元矩阵方程的拆分,得到算子分裂子矩阵组,进而使用具有一阶和二阶精度的算子分裂法对FP方程进行数值求解。针对线性系统和非线性Duffing系统进行了FP方程数值求解的验证,检验了拆分为对流项和扩散项算子的计算精度和计算时间。实验结果表明,相对于传统的Runge-Kutta求解方法,在相同的数值解精度下,结合算子分裂法的求解时间仅为纯有限单元法的1%~5%。有限单元法结合算子分裂是一种具有较快速计算潜力的FP方程数值求解方法。

An efficient approach for the equivalent linearization of frame structures with plastic hinges under nonstationary seismic excitations

An efficient approach is proposed for the equivalent linearization of frame structures with plastic hinges under nonstationary seismic excitations.The concentrated plastic hinges,described by the Bouc-Wen model,are assumed to occur at the two ends of a linear-elastic beam element.The auxiliary differential equations governing the plastic rotational displacements and their corresponding hysteretic displacements are replaced with linearized differential equations.Then,the two sets of equations of motion for the original nonlinear system can be reduced to an expanded-order equivalent linearized equation of motion for equivalent linear systems.To solve the equation of motion for equivalent linear systems,the nonstationary random vibration analysis is carried out based on the explicit time-domain method with high efficiency.Finally,the proposed treatment method for initial values of equivalent parameters is investigated in conjunction with parallel computing technology,which provides a new way of obtaining the equivalent linear systems at different time instants.Based on the explicit time-domain method,the key responses of interest of the converged equivalent linear system can be calculated through dimension reduction analysis with high efficiency.Numerical examples indicate that the proposed approach has high computational efficiency,and shows good applicability to weak nonlinear and medium-intensity nonlinear systems.

含几何缺陷旋转FG-CNTRC梁非线性自由振动分析

以含有初始几何缺陷的功能梯度碳纳米管增强旋转梁的非线性自由振动行为为对象开展研究。根据Euler–Bernoulli梁理论,引入von Kármán几何非线性假设,利用Hamilton原理,推导出系统的非线性偏微分控制方程。随后,采用Galerkin法,将控制方程转化为常微分方程,并利用变分迭代法,求得系统非线性频率的解析解。最后,通过数值结果,讨论材料属性、转速、轮毂半径和几何缺陷等因素对结构非线性频率的影响。

粘滞阻尼器减震结构非线性随机振动的时域显式降维迭代随机模拟法

粘滞阻尼器在大型复杂结构减震设计中应用广泛。由于粘滞阻尼器的非线性阻尼力特性,粘滞阻尼器减震结构非平稳随机地震反应分析是一个典型的局部非线性随机振动问题。利用减震结构动力响应时域显式表达式的降维列式优势,仅针对与粘滞阻尼器相关的局部自由度进行非线性迭代计算,提出了局部非线性随机振动问题的时域显式降维迭代随机模拟法,为设置粘滞阻尼器的大型复杂减震结构非线性地震反应分析提供一种高效的随机振动方法。以安装了四个纵桥向粘滞阻尼器的某主跨1200m悬索桥为工程实例,开展E2水准地震激励下的非线性随机振动分析。计算结果显示,设置阻尼器后,主梁的纵桥向位移得到明显控制,降幅达到80%,大桥的关键截面内力也有5%左右的降幅。

大规模非线性系统随机振动显式迭代Monte Carlo模拟法

探讨了非平稳随机激励下大规模非线性系统随机振动Monte Carlo模拟法。引入等效激励的概念,把非线性系统动力方程写成状态方程的形式,并采用精细积分法对状态方程进行数值求解,导出非线性系统振动分析的显式迭代法。基于所导出的显式迭代公式,可以有效提高单次确定性非线性振动分析的计算效率,从而可以通过Monte Carlo模拟获得非平稳随机激励下非线性系统随机响应的统计信息,同时还可以获取非线性系统随机响应的演化概率密度函数。数值算例表明,所提出的方法迭代收敛速度快,计算精度高,适用于求解大规模非线性系统随机振动问题。

非平稳随机响应灵敏度分析的时域显式法

针对非平稳激励下的结构振动问题,研究动力响应方差灵敏度的高效时域求解算法。首先推导确定性动力响应灵敏度的时域显式表达,继而结合方差灵敏度的一般计算公式,得到非平稳随机响应方差灵敏度的时域显式计算列式。该列式同样适用于平稳激励下结构瞬态响应方差灵敏度的求解。以框架结构和桁架结构分别受不同类型的非平稳激励为例,时域显示解法和其他方法的对比计算结果验证了时域显示解法的有效性。

考虑土体非线性的部分埋入群桩竖向振动分析

在桩基振动问题中,桩侧土往往表现出显著的非线性特性,土体动模量和阻尼的大小与桩身位移相关.目前的分析方法一般只采取恒定的土性参数,忽略了非线性的影响.基于部分埋入群桩竖向振动的研究成果,以桩身位移与土性参数之间的相互关系为依据,运用迭代方法考虑了土体非线性对部分埋入群桩振动的影响,分析了迭代收敛的条件,并通过算例分析了非线性的影响程度,观察了它与荷载大小的关系.最后,将该方法应用于分析某海上风力发电机组群桩基础的振动特性,得到其阻抗值及位移响应曲线.

双索悬索桥结构参数对自振特性的影响分析

引入只受拉三维拉索单元,采用考虑几何非线性的子空间迭代法对黄河大峡水库下游某双索悬索桥自振特性进行分析,理论值与实测值能较好的吻合,说明该空间非线性有限元分析方法的正确性;进而将该桥与相同跨径和结构参数的单索悬索桥的自振频率、振型进行对比分析,结果表明双索悬索桥能有效提高桥梁一阶竖弯振动频率。最后,讨论恒载集度、加劲梁刚度、矢跨比等结构参数变化对双索悬索桥自振特性的影响,为双索悬索桥结构设计理论提供了动力性能方面的依据。

振动力学发展历史概述

概述振动力学的发展历史．着重分析振动力学几个主要方面的演化过程，包括线性振动理论，振动问题近似解法，非线性振动，以及随机振动．

被动隔振体非线性振动的能量迭代解法

研究了由基础振动激励、弹性材料隔离的被动隔振体的强非线性动力响应。用变形的三次多项式函数表征隔振材料的非线性刚度特性,建立了被动隔振体的非线性动力学方程,得到有阻尼受迫振动Duffing方程。将求解强非线性自治系统的能量迭代方法加以改进,推广应用到强非线性非自治系统,求出周期响应的近似解析解表达式,以及幅频关系、相频关系和隔振系数的近似表达式。算例中应用本方法与Runge-Kutta方法进行了对照,结果表明求解精度较高。本文利用计算机进行了辅助推导。

强非线性主动隔振系统的运动响应及传递率

建立了主动隔振体的非线性动力学方程,即有阻尼受迫振动Duffing方程;对求解强非线性自治系统的能量迭代方法加以改进,将其用于求解强非线性非自治系统,得到了主动隔振系统周期运动响应的解析表达式和振幅—频率关系曲线,并按新振动传递率定义研究了振动传递率与频率的关系。应用这一方法,获得了精度较高的周期解表达式、振幅与频率关系曲线以及位移传递率与频率关系曲线;得到了主动隔振问题的有关结果对于非线性硬弹簧系统(α>0),随着非线性项系数增大,共振的振幅虽然减小,但传递率增大,故隔振效果较差;对于非线性软弹簧系统(α≤0),随着非线性项系数的绝对值增大,共振的振幅减小,同时传递率也减小,故非线性软弹簧系统(α≤0)具有较好的主动隔振。

基于等效线性化法的弱非线性体系下的地震响应分析

对弱非线性单自由度非自由振动系统结构在白噪声地震激励下的随机响应问题进行了研究。首先,建立了结构的弱非线性响应方程;然后根据能量平衡原理对运动方程进行了等效线性化;最后获得了金井清谱(Kanai-Tajimi谱)下等效线性化方程的位移响应及速度响应的方差,将弱非线性随机响应问题简化为求解线性随机响应问题,并给出了算例验证。结果表明,该方法可以准确得到弱非线性随机响应问题的解。同时,建立了单自由度结构弱非线性随机响应频域分析的统一解析解法。

用时域仿真求解一随机振荡系统的水准跨越率

要为获得一在随机激励下的非线性结构系统的响应的向上水准跨越率而进行了蒙特卡罗仿真。该结构系统的非线性包括Duffing刚度和考虑海流存在时的Morison拖力,四阶Runge-Kutta法被用来对该结构系统在不同海流速度下的运动微分方程进行数值积分,得到了响应值的一阶导数以及响应值与响应值一阶导数的联合概率密度,接下来利用著名的R ice公式计算了系统响应的向上水准跨越率。这是首次对该类型非线性系统进行数值仿真,并且获得了系统响应的水准跨越率的精确预报。该跨越率值是求取许多有关响应统计和可靠性分析的重要数据的关键。

正交异性扁薄球壳的非线性轴对称振动

给出了一种研究圆柱正交异性扁薄球壳非线性轴对称自由振动的新的时间模态方法。首先,使用变分法导出了决定振动频率的一对代数-微分方程。然后,利用作者提出的改进的修正迭代法求得了渐进解。这使此种壳体的非线性振动的进展获得扩充研究。