Exact solutions for longitudinal vibration of rods coupled by translational springs

The objective of this paper is to present exact analytical solutions for longitudinal vibration of non-uniform rods with concentrated masses coupled by translational springs. Using appropriate transformation, the governing differential equation for longitudinal vibration of a rod with varying cross section is reduced to Bessel's equation or an ordinary differential equation with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the area variation. The exact solutions for free longitudinal vibration of rods with varying cross-section are derived. The initial parameter method and the transfer matrix method are proposed to establish the frequency equation for the longitudinal vibration of two rods coupled by translational springs. The advantage of the proposed methods is that the frequency equation for two rods coupled by translational springs can be established in terms of a determinant of 2-order for any number of translational springs and concentrated masses. The proposed methods can be used to solve the problem of free longitudinal vibration of uniform and non-uniform rods with concentrated masses coupled by various translational springs, and thus to investigate the axial stiffness and mass distribution among the rods to alter the system's dynamic characteristics. A numerical example shows that the fundamental longitudinal natural frequency of two reaction towers coupled by a pipe calculated by the proposed methods is in good agreement with the full scale measured data, suggesting the proposed methods are applicable to engineering practices.     

恒速状态移动质量作用的柔性梁最优振动控制

针对恒速状态移动质量作用下柔性梁的最优振动控制方案进行研究分析。根据柔性梁振动理论,考虑移动质量与柔性梁弹性振动之间的耦合作用,建立移动质量-柔性梁时变系统的振动方程。当移动质量刚好离开梁时给出终端时刻点,超越终端时刻的梁作自动振动,建立时不变系统的振动方程。若将以往适用时不变系统的最优控制方法应用于时变系统,会导致次优化效果。为此,采用多种控制设计方法对系统的振动响应进行抑制,讨论不同控制方案的优缺性。数值结果表明:对于特定的作动器位置,有必要采用具有时变特征的控制方法替代时不变控制方法;相比时变控制方法,采用状态相关的黎卡提方程(state-dependent riccati equation,简称SDRE)控制方法得到的性能指标最优;基于增广矩阵的线性二次型调节器(linear quadratic regulator,简称LQR)边界控制效果优于bang-bang控制,前者控制策略得到的性能指标最优。     

移动最小二乘法的时变结构模态参数辨识

研究时变结构模态参数辨识,基于泛函矢量时变自回归模型(Functional series vector time-dependent AR model,FS-VTAR)提出一种改进的移动最小二乘法的时变结构模态参数辨识方法。该方法源于无网格法中构造形函数进行局部近似的思想,引入带权正交基函数对移动最小二乘(Moving least square,MLS)的基函数进行改进,使得在辨识时间域内构造形函数矩阵过程中不再出现数值条件问题,从而提高了计算精度。把时变系数在形函数上线性展开,利用最小二乘法得到形函数的系数,从而得到时变系数。把时变模型特征方程转换为广义特征值问题提取出模态参数。利用时变刚度系统非平稳振动信号验证该方法,结果表明:改进的移动最小二乘法相比于传统的FS-VTAR模型能有效地避免基函数形式的选择和很高的基函数阶数且更加高效,相比于移动最小二乘法能有效地避免辨识过程中的数值问题,具有更高的模态参数辨识精度。     

Exact dynamic stiffness matrix of a Timoshenko three-beam system

An exact dynamic stiffness matrix is established for an elastically connected three-beam system, which is composed of three parallel beams of uniform properties with uniformly distributed-connecting springs among them. The formulation includes the effects of shear deformation and rotary inertia of the beams. The dynamic stiffness matrix is derived by rigorous use of the analytical solutions of the governing differential equations of motion of the three-beam system in free vibration. The use of the dynamic stiffness matrix to study the free vibration characteristics of the three-beam system is demonstrated by applying the Muller root search algorithm. Numerical results for the natural frequencies and mode shapes of the illustrative examples are discussed for 10 interesting boundary conditions and three different stiffness constants of springs.     

Influence of contact wire pre-sag on the dynamics of pantograph-railway catenary

Stiffness variations and wave propagation in railway catenaries are the main sources of contact loss between the pantograph and the railway catenary. The pre-sag of a contact wire is sometimes introduced to compensate for the stiffness variation. To include the pre-sag’s influence on the dynamic interaction of the pantograph-catenary, we present a modified single-degree-of-freedom (SDOF) dynamic system with a time-varying stiffness that is based on a simplified model from a previous study. The pantograph-catenary parameters that are related to the optimal pre-sag to minimise variations in the contact force are obtained by solving the equations of the SDOF dynamic system. The amount of optimal pre-sag is determined from parameters such as speed, span length, uplift, and the stiffness variation coefficient. Because wave propagation is not allowed in the SDOF model, we investigate the influence of pre-sag on the dynamic interaction between the pantograph and the catenary by using both SDOF and finite element method (FEM) models. These models show that, by using the optimal pre-sag, we can significantly reduce the variation in the contact force in the medium-speed range, which is near the 1/2 sub-resonance speed. The influence of pre-sag is confirmed by a field test that was performed on a Korean conventional catenary. However, the FEM model shows that in the high-speed range near 300 km h⁻¹, the introduction of the pre-sag can reduce the variation in the contact force slightly only if uplift force subject to the pantograph is large enough to cause slackening in the droppers next to the supports. Otherwise, it does not reduce the variation in the contact force. The FEM also shows that the optimal pre-sag (less than 1/2000 of span length) in the high-speed range is much smaller than that (about 1/1000 of span length) in the medium-speed range.     

多自由度无阻尼动力吸振器的理论研究

讨论了将简谐激振下无阻尼单自由度主系统的动力消振原理推广到多频谐波激励下无阻尼多自由度主系统的情况,通过选择一个合适的多自由度无阻尼动力吸振从系统。即从系统的固定频率正好是所有的激振力的频率,可以完全消除多频谐波激励下无阻尼多自由度主系统中某质点的振动响应。还讨论了根据给定弹簧-质量系统的特征值及对该系统进行小的、已知的修正后所得系统的特征值,来直接重构质量矩阵和刚度矩阵的方法。     

纯电动汽车高速斜齿轮传动振动特性分析

为研究高转速情况下时变啮合刚度和啮合冲击对斜齿轮传动振动特性的影响,以某纯电动汽车高速斜齿轮传动为研究对象,建立了弯-扭-轴动力学模型;采用改进的基于承载接触分析的计算方法获得时变啮合刚度曲线,并计算了啮合冲击时间及啮合冲击力幅值;分析了时变啮合刚度、啮合冲击以及两者综合3种激励条件下高速斜齿轮传动系统的振动特性。结果表明：时变啮合刚度激励下,在过共振区,转速变化对系统振动的影响不显著;啮合冲击激励以及综合激励条件下,系统振动随转速的升高而增大,与啮合冲击激励相比,综合激励下振动加速度增幅较缓。研究结果可为纯电动汽车高速斜齿轮传动的设计和工程应用提供参考依据。     

数控机床时变切削振动自适应稳定控制方法

针对数控机床切削过程中因不确定因素多,导致刀尖运动刚度和阻尼力难以保持恒定而造成的强振动问题,提出一种基于模态稳定的自适应稳定控制方法。模拟机床切削的连贯加工过程,获得影响最大的阻尼力和刚度系数;将其作为模糊动态振动表达模型的初始输入数值,通过响应函数得到二者在时变振动状态下的分布可能结果;利用置信函数得到阻尼和刚度系数的安全置信区间,以该区间作为判定依据,对超过区间的系数实行模态控制,完成切削振动的稳定控制。从切削振动的位移信号和声音信号 2 个方面进行仿真验,结果表明,该方法具有一定的可靠性和稳定性,可保证刚度和阻尼系数的状态恒定。     

圆柱滚子轴承弹流接触副刚度及阻尼系数研究

针对圆柱滚子轴承中滚子-滚道弹流接触副,建立有限长线接触非稳态弹流润滑模型,利用追赶法、快速傅里叶变换和Newmark技术数值求解接触副在自由振动下的衰减曲线。以刚体接近距离的变化作为判断接触副振动的标准,结合弹流润滑模型和有阻尼系统的自由振动模型,给出预测滚子-滚道弹流接触副动力学参数的方法,考察初始扰动量、润滑剂黏压系数、滚子长度和载荷对刚度和阻尼系数的影响。结果表明：小扰动下,初始扰动量大小对刚度和阻尼系数的影响可忽略不计;弹流润滑下的刚度小于干接触下的Hertz接触刚度;增加润滑剂黏压系数、滚子长径比和载荷,均可增大弹流接触副的刚度系数;阻尼系数随润滑剂黏压系数的增加而减小,随滚子长径比的增加而增加,随一定范围内载荷的增加而减小。     

An exact approach for free vibration analysis of rectangular plates with line-concentrated mass and elastic line-support

An exact approach for free vibration of an isotropic rectangular plate carrying a line-concentrated mass and with a line-translational spring support or carrying a line-spring-mass system is presented in this paper. The mode shape function of vibration of such a plate is expressed in terms of the four fundamental solutions derived in this paper. The main advantage of the proposed method is that the resulted frequency equation for such a rectangular plate can be conveniently obtained from a second-order determinant. The proposed method is thus computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures which usually led to an eighth-order determinant for solving the title problem. Two numerical examples are given to illustrate the efficiency of the proposed method and to investigate the effects of the location and the magnitude of a line-concentrated mass and elastic line-support as well as the influence of the aspect ratio on the natural frequencies of a rectangular plate.     

基于指数窗截取递推最小二乘法的齿轮啮合刚度辨识算法研究

齿轮在机械传动系统中有着广泛应用,由于齿轮啮合过程中参与啮合的轮齿对数周期变化,因此,齿轮啮合刚度为时变参数,在啮合时会产生啮合振动。当齿轮副出现齿根裂纹时,啮合刚度会减小,齿轮啮合产生的系统振动响应也发生改变,通过振动响应辨识齿轮啮合刚度能够监测齿轮副的健康状态。针对齿轮啮合刚度的时变特征,提出了基于指数窗截取递推最小二乘(Exponential window recursive least square,EWRLS)算法和振动信号瞬时频率的齿轮啮合刚度辨识方法。进行啮合刚度辨识时,EWRLS算法将输入、输出齿轮的转速曲线分别作为辨识输入信号和观测信号,使用指数窗函数进行数据截断,使用递推最小二乘算法估计系统参数。为了计算输入、输出齿轮的转速曲线,使用经验模态分解(Empirical mode decomposition,EMD)方法将振动信号分解为具有不同变化频率的本征模态函数(Intrinsic mode function,IMF),并根据IMF的平均频率重构输入、输出齿轮的特征信号。通过Hilbert变换计算特征信号的瞬时频率曲线,从而获得各齿轮的转速曲线。使用仿真和实测信号对算法进行验证,结果表明,EWRLS算法能够辨识齿轮副的时变啮合刚度。     

Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations

This paper is concerned with the vibration behaviour of rectangular Mindlin plates resting on non-homogenous elastic foundations. A rectangular plate is assumed to rest on a non-homogenous elastic foundation that consists of multi-segment Winkler-type elastic foundations. Two parallel edges of the plate are assumed to be simply supported and the two remaining edges may have any combinations of free, simply supported or clamped conditions. The plate is first divided into subdomains along the interfaces of the multi-segment foundations. The Levy solution approach associated with the state space technique is employed to derive the analytical solutions for each subdomain. The domain decomposition method is used to cater for the continuity and equilibrium conditions at the interfaces of the subdomains. First-known exact solutions for vibration of rectangular Mindlin plates on a non-homogenous elastic foundation are obtained. The vibration of square Mindlin plates partially resting on an elastic foundation is studied in detail. The influence of the foundation stiffness parameter, the foundation length ratio and the plate thickness ratio on the frequency parameters of square Mindlin plates is discussed. The exact vibration solutions presented in this paper may be used as benchmarks for researchers to check their numerical methods for such a plate vibration problem. The results are also important for engineers to design plates supported by multi-segment elastic foundations.     

Vibration analysis of thin-walled curved beams using DQM

The differential quadrature method (DQM) is applied to computation of the eigenvalues of small amplitude free vibration for thin-walled curved beams including a warping contribution. Natural frequencies are calculated for singlespan, curved, wide-flange uniform beams having a range of nondimensional parameters representing variations in warping stiffness, torsional stiffness, radius of curvature, included angle of the curve, polar mass moment of inertia, and various end conditions. Results are compared with existing exact and numerical solutions by other methods for cases in which they are available. It is found that the DQM gives good accuracy even when only a limited number of grid points is used. In addition, results are given for a cantilever beam not previously considered for this problem. Finally, parametric results are presented in dimensionless form.     

Model updating of damped structures using FRF data

Due to the important contribution of damping on structural vibration, model updating of damped structures becomes significant and remains an issue in most model updating methods developed to date. In this paper, the frequency response function(FRF) method, which is one of the most frequently referenced model updating methods, has been further developed to identify damping matrices of structural systems, as well as mass and stiffness matrices. In order to overcome the problem of complexity of measured FRF and modal data, complex updating formulations using FRF data to identify damping coefficients have been established for the cases of proportional damping and general non-proportional damping. To demonstrate the effectiveness of the proposed complex FRF updating method, numerical simulations based on the GARTEUR structure with structural damping have been presented. The updated results have shown that the complex FRF updating method can be used to derive accurate updated mass and stiffness modelling errors and system damping matrices.     

Note: real time control of a tunable vibration absorber based on magnetorheological elastomer for suppressing tonal vibrations

An adaptive tunable vibration absober based on magnetorheological elastomer (MRE) is designed as an intelligent device for auto-tuning itself to the time-varying harmonic disturbance force to reduce the unwanted vibration of the primary system in the steady state. The objectives of this note are to develop and implement a continuous control method for a MRE tunable vibration absorber (TVA) and to evaluate its performance in suppressing time-varying tonal vibrations. In the proposed control, the stiffness of MREs is continuously varied based on a nonlinear tuning function that relates the response of the system to the input magnetic field density. Through experiments, it will be shown that the proposed MRE TVA reduces in real time the transmission of a time-varying excited vibration of 48-55 Hz, which shows the potential applicability of the MRE in reducing unwanted vibration to precision devices.     

瞬态激励、应变片测量、非线性最优化识别振动系统模态参数

本文提出了用力锤激励、应变片测量响应、用非线最优化方法来识别振动系统模态参数的理论和方法。本文提出了“撼应系数”这一概念。通过撼应系数及柔度系数和刚度系数,找到了应变和位移的关系。文中导出了用应变片测量响应时系统的应变导纳函数公式。采用本文所述方法可识别出振动系数的固有频率、位移振型、应变振型、应力振型、广义阻尼及主刚度和主质量等模态参数。     

Contact finite element method for dynamic meshing characteristics analysis of continuous engaged gear drives

The dynamic meshing characteristics of gear drives have been a major concern in the design of power transmission systems as they affect vibration, acoustic noise, durability and efficiency. Gaining a more comprehensive understanding of the dynamic meshing characteristics of continuous engaged gear drives is a key to the development of power transmission systems. In this paper, a dynamic contact finite element analysis method, considering the variation of the engaged teeth pairs, the loaded elastic and contact deformations, and the sliding friction, is presented for the dynamic meshing characteristics analysis of continuous and elastic engaged gear drives. Various kinds of continuous engaged gear models under low and high speed condition are simulated and compared using the presented method. The tooth profile modification was designed based on the simulation results. Moreover, the effects of the tooth profile modification, the sliding friction and the time-varying meshing stiffness upon the dynamic meshing characteristics of continuous engaged gear drives are discussed in detail. The results show that the method is not only effective in designing and evaluating the tooth profile modification, but also in studying the dynamic meshing characteristics of continuous engaged gear drives with realistic time-varying meshing stiffness and tooth sliding friction. The present method could provide an effective tool for vibration mechanism study and dynamic design of the continuous engaged gear drives considering more influence factors.     

含间隙和时变刚度的齿轮机构振动特性分析及实验研究

以齿轮系统动力学和非线性动力学理论为基础,针对齿轮机构时变啮合刚度和齿侧间隙耦合作用的具体特点,并考虑油膜挤压刚度,提出了基于轮齿弹性振动及单、双齿啮合区变化的齿轮机构多体弹性非线性动力学模型,用GEAR方法对其在某些参数域中进行了非线性振动研究,得出不同的参数对齿轮机构动力学特性的影响,并提出了实验方案。     

考虑时变刚度的齿轮振动响应仿真分析

利用MASTA软件建立齿轮传动系统实体模型,并在MASTA环境下调用ANSYS软件完成齿轮传动系统刚度有限元分析,计算得到由齿轮时变啮合刚度引起的传动误差,以此作为齿轮箱体响应分析的激励源,利用ANSYS软件对箱体进行振动响应分析,分析了时变刚度对箱体振动响应的影响.本文所采用的方法,可用于实际工程中分析预估齿轮传动装...     

Vibratory characteristics of flexural non-uniform Euler–Bernoulli beams carrying an arbitrary number of spring–mass systems

A new exact method for the analysis of free flexural vibrations of non-uniform multi-step Euler–Bernoulli beams carrying an arbitrary number of single-degree-of-freedom and two-degree-of-freedom spring–mass systems is presented in this paper. The closed-form solutions for free vibrations of non-uniform Euler–Bernoulli beams are derived for five important cases. Then, using the massless equivalent springs to replace the spring–mass systems and the fundamental solutions developed in this paper, the frequency equation for free flexural vibrations of a multi-step non-uniform beam with any kind of support configurations and carrying an arbitrary number of spring–mass systems can be conveniently established from a second-order determinant. The proposed method is computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures.