A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect

A single-walled nanotube structure embedded in an elastic matrix is simulated by the nonlocal Euler-Bernoulli, Timoshenko, and higher order beams. The beams are assumed to be elastically supported and attached to continuous lateral and rotational springs to take into account the effects of the surrounding matrix. The discrete equations of motion associated with free transverse vibration of each model are established in the context of the nonlocal continuum mechanics of Eringen using Hamilton's principle and an efficient meshless method. The effects of slenderness ratio of the nanotube, small scale effect parameter, initial axial force and the stiffness of the surrounding matrix on the natural frequencies of various beam models are investigated for different boundary conditions. The capabilities of the proposed nonlocal beam models in capturing the natural frequencies of the nanotube are also addressed.     

Free vibration analysis of straight-line beam regarded as distributed system by combining Wittrick-Williams algorithm and transfer dynamic stiffness coefficient method

We developed a method for analyzing the free vibration of a structure regarded as a distributed system, by combining the Wittrick-Williams algorithm and the transfer dynamic stiffness coefficient method. A computational algorithm was formulated for analyzing the free vibration of a straight-line beam regarded as a distributed system, to explain the concept of the developed method. To verify the effectiveness of the developed method, the natural frequencies of straight-line beams were computed using the finite element method, transfer matrix method, transfer dynamic stiffness coefficient method, the exact solution, and the developed method. By comparing the computational results of the developed method with those of the other methods, we confirmed that the developed method exhibited superior performance over the other methods in terms of computational accuracy, cost and user convenience.     

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.     

Dynamic finite element method for generally laminated composite beams

A dynamic finite element method for free vibration analysis of generally laminated composite beams is introduced on the basis of first-order shear deformation theory. The influences of Poisson effect, couplings among extensional, bending and torsional deformations, shear deformation and rotary inertia are incorporated in the formulation. The dynamic stiffness matrix is formulated based on the exact solutions of the differential equations of motion governing the free vibration of generally laminated composite beam. The effects of Poisson effect, material anisotropy, slender ratio, shear deformation and boundary condition on the natural frequencies of the composite beams are studied in detail by particular carefully selected examples. The numerical results of natural frequencies and mode shapes are presented and, whenever possible, compared to those previously published solutions in order to demonstrate the correctness and accuracy of the present method.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

Use of a generalized beam/spring element to analyze natural vibration of rail track and its application

This paper describes the formulation of a generalized beam/spring track element to obtain the natural vibration characteristics of a railway track modeled as a periodic elastically coupled beam system on a Winkler foundation. The rail/tie beams are described by either the Timoshenko beam theory or the Bernoulli-Euler beam theory. The rail beam is assumed to be discretely coupled to the cross-track ties through the coupling spring elements at the periodic rail/tie intersections. The generalized beam/spring element consists of a rail span beam segment, two adjacent tie beams, the coupling spring elements and the ultimate foundation stiffness. The entire track/beam system is then discretized into an assembly of periodic structural units. An equivalent frequency-dependent spring coefficient representing the resilient, flexural and inertial characteristics of the track substructure unit is formulated to establish the dynamic stiffness matrix of the generalized element. The eigenvalue problem of the track/beam system is solved by employing a comprehensive and efficient numerical routine. Solutions are provided for the natural frequencies of the track and the mode shapes of the rail/tie beams under transversely (cross-track) symmetric vibration. The natural vibration results are used to obtain the dynamic receptance response of a typical field track and to compare them with an existing model and field experimental data.     

Dynamic responses of composite H-beams with different elastic couplings

Governing equations are derived via Hamilton’s principle for composite thin-walled H-type open cross-section beams that show a number of non-classical effects such as transverse shear, primary and secondary warping, and anisotropy of constituent materials. The vibration characteristics of composite thin-walled beams with different elastic couplings, such as circumferentially asymmetric stiffness (CAS) and circumferentially uniform stiffness (CUS) configurations are investigated with respect to the bending-transverse shear coupling and the bending-twist coupling resulting from the directional properties of fiber reinforced composite materials. The dynamic responses of anisotropic thin-walled beams to harmonic and exponentially time-dependent loads are also investigated. It was revealed that transverse shear, elastic couplings, and warping effects greatly influence the free vibration and dynamic response characteristics of composite H-type open cross-section beams.     

康复机器人准零刚度隔振特性分析与实验

为了解决康复机器人低频振动对人体的影响问题,提出并设计一种具有准零刚度特性的新型被动隔振机构,它由两个对称的具有负刚度特性的拉伸弹簧并联一个线性正刚度弹簧实现。首先,通过静力学特性,建立此隔振机构的力-位移和刚度-位移关系式,得出机构在静态平衡位置处具有准零刚度特性的参数条件;其次,通过动力学特性,建立其分别在简谐力和简谐位移激励下的非线性微分动力学方程,采用谐波平衡法分析机构参数与激励对系统力传递率、位移传递率的影响;最后,通过实验验证在有、无负刚度情况下,输入与输出的曲线。结果表明,在有负刚度机构的情况下,隔振系统具有一定范围的准零刚度特性,且低频隔振性能较好,达到了高静态、低动态刚度的效果。本研究对机器人低频隔振具有创新性和指导意义。     

Free vibration analysis of sandwich beam carrying sprung masses

In this paper, free vibration of three-layered symmetric sandwich beam carrying sprung masses is investigated using the dynamic stiffness method and the finite element formulation. First the governing partial differential equations of motion for one element are derived using Hamilton’s principle. Closed form analytical solution of these equations is determined. Applying the effect of sprung masses by replacing each sprung mass with an effective spring on the boundary condition of the element, the element dynamic stiffness matrix is developed. These matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the well known Wittrick–Williams algorithm. Free vibration analysis using the finite element method is carried out by increasing one degree of freedom for each sprung mass. Finally, some numerical examples are discussed using the dynamic stiffness method and the finite element formulation. After verification of the present model, the effect of various parameters such as mass and stiffness of the sprung mass is studied on the natural frequencies.     

汽车起重机振动模态研究

汽车起重机在起吊工作过程中,激励频率应远离固有频率,以免因产生共振而发生危险。因此在汽车起重机设计与应用过程中对其进行振动模态分析是十分必要和重要的工作。将汽车起重机处理为由多个刚体、集中质量、弹性梁、扭簧以及弹簧按一定方式铰接而成的刚柔耦合多体系统,建立了汽车起重机的动力学模型,用多体系统传递矩阵法计算了其振动固有频率及对应的主振型,解决了同时含有多个刚体和弹性体的复杂多体系统的振动模态计算问题。     

考虑翘曲影响的Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀薄壁梁单元弯扭耦合振动的运动偏微分方程，推导其自由振动时的精确动态传递矩阵。采用考虑翘曲影响的Bernoulli-Euler梁理论，且假定薄壁梁单元的横截面是单对称的。动态传递矩阵可以用于计算薄壁梁集合体的精确固有频率和模态形状。针对两个薄壁梁算例，采用自动Muller法和结合频率扫描法的二分法求解频率特征方程，并讨论翘曲刚度对弯扭耦合：Bernoulli-Euler薄壁梁固有频率的影响。数值结果验证了本文方法的精确性和有效性，并指出翘曲刚度可以显著改变薄壁开口截面梁的固有频率。     

An efficient numerical method for predicting the natural frequencies of cylindrical helical springs

In this study, the stiffness method is employed for the free vibration problem of cylindrical helical springs. The element stiffness matrix for the helical spring with twelve degrees-of-freedom is obtained exactly by the transfer matrix method. The efficacious numerical algorithm is employed for the computation of the element transfer matrix. The concentrated element mass matrix is used. The subspace iteration method is preferred for the solution of the large-scale eigenvalue problem. The axial and shear deformation and the rotary inertia terms are considered in the formulation. The free vibrational parameters are chosen as the number of coils (n=3–16), the helix pitch angle (α=5–25°), the shape of cross-section (circular, hollow circle and squared) and as the ratio of the diameters of cylinder to wire (D/d=4–16) in a wide range. Solving the miscellaneous problems, the non-dimensional charts are obtained for the cylindrical helical springs fixed at both ends. Using these charts the natural frequencies are expressed in analytical form in a very good approximation (with the maximum absolute relative error of 5%) and presented for the designers.     

气膜端面密封角向摆动自振稳定性

提出造成气膜端面密封角向摆动自振的内在机理，既有半频摆动自振又有角向气锤自振。在其稳定性分析中给出了不同端面结构、浮环支撑弹簧和二次密封阻尼等的影响。分析按照小扰动线性化的分布参数法，联立气膜微扰雷诺方程和浮环微扰运动方程，对密封系统的角向摆动自振稳定性界限进行了数值迭代。     

Development of free vibration analysis algorithm for beam structures by combining Sylvester’s inertia theorem and transfer stiffness coefficient method

A new free vibration analysis method, which is called the Sylvester-transfer stiffness coefficient method (S-TSCM), is developed by combining Sylvester’s inertia theorem and the transfer stiffness coefficient method. In this paper, the free vibration analysis algorithm of a straight-line beam structure is formulated by S-TSCM. From the computation results of the free vibration analysis for the three types of beam structures, we confirm that S-TSCM is a very effective method. In particular, S-TSCM is superior to both the transfer stiffness coefficient method and the transfer matrix method in terms of computational accuracy and time. In the free vibration analysis for the beam structure with a large number of degrees-of-freedom, S-TSCM is superior to the finite element method in terms of computational time and storage.     

轴承载荷对转子系统各支承的耦合振动分析

以多支承转子 轴承 弹性基础系统为研究对象,对多支承转子系统某支承处轴承载荷变化引起不同支承点处轴及轴承的耦合振动响应进行了研究。首先,建立了多支承转子系统支承间的耦合振动模型,并进行了仿真和试验,耦合振动模型考虑了每个支承处的油膜耦合效应,通过轴承载荷敏感度矩阵得到各支承耦合力;然后,利用龙格库塔法对油膜刚度为定刚度和动刚度两种情况下的转子系统进行了数值分析及仿真;最后,在8支承转子试验台上进行试验,采取改变某一支承处的位移来得到轴承载荷的变化,并提取各支承处的振动信号。结果表明,轴承位置在稳态和瞬态两种情况下改变时振动响应明显不同,油膜刚度为动刚度的分析结果更为合理。     

An exact solution for the vibration of helical springs using a Bernoulli-Euler model

A solution is obtained for the steady-state vibration behaviour of uniform helical springs in which the following effects are ignored; deformation due to shear and axial loads, the rotational inertias of the cross-section, and static loads applied to the spring. A theory is developed for the evaluation of the dynamic stiffness matrix. Natural frequencies are found using an adaption of the Wittrick-Williams algorithm. Details of the method of calculation are discussed. Results are presented for natural frequencies, and are compared with values obtained using other methods and assumptions. The method of this paper is shown to be particularly efficient.     

A tunable dynamic vibration absorber for unbalanced rotor system

A tunable dynamic vibration absorber for unbalanced rotor system which is made up of coil springs and magnetic spring is presented. The structure of the absorber is introduced and the stiffness tuning mechanism of the magnetic spring is explained. A finite element model of the rotor-absorber system was built and the influencing factors to the appearance of the absorber were studied numerically. Finally, experiments were carried out to verify the numerical results, and PID control strategy was tested. The numerical and experimental results show that the present absorber is effective for vibration suppression of an unbalanced rotor system, and the control strategy is effective.     

Bending and vibration of a discontinuous beam with a curvic coupling under different axial forces

The transverse stiffness and vibration characteristics of discontinuous beams can significantly differ from those of continuous beams given that an abrupt change in stiffness may occur at the interface of the former. In this study, the equations for the deflection curve and vibration frequencies of a simply supported discontinuous beam under axial loads are derived analytically on the basis of boundary, continuity, and deformation compatibility conditions by using equivalent spring models. The equation for the deflection curve is solved using undetermined coefficient methods. The normal function of the transverse vibration equation is obtained by separating variables. The differential equations for the beam that consider moments of inertia, shearing effects, and gyroscopic moments are investigated using the transfer matrix method. The deflection and vibration frequencies of the discontinuous beam are studied under different axial loads and connection spring stiffness. Results show that deflection decreases and vibration frequencies increase exponentially with increasing connection spring stiffness. Moreover, both variables remain steady when connection spring stiffness reaches a considerable value. Lastly, an experimental study is conducted to investigate the vibration characteristics of a discontinuous beam with a curvic coupling, and the results exhibit a good match with the proposed model.

     

纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

基于结构动态特性的磁力轴承转子系统的设计

分析了径向磁力轴承动力学特性,计算了磁力轴承的刚度系数和阻尼系数。在此基础上,对磁力轴承的转子结构动态特性进行了有限元分析。提出了基于结构动态特性的磁力轴承转子控制系统的设计方法。