An investigation on effects of traveling mass with variable velocity on nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions

In this paper, the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a traveling mass with variable velocity is investigated. The nonlinear coupled partial differential equations of motion for the bending rotation of cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. These nonlinear coupled PDEs are solved by applying Galerkin’s method to obtain dynamic response of the beam under the act of a moving mass. The appropriate parametric studies by taking into account the effects of the magnitude of the traveling mass, the velocity of the traveling mass with a constant acceleration/deceleration and effect of different beam’s boundary conditions are carried out. The beams’ large deflection has been captured by including the stretching effect of its mid-surface. It was seen that the existence of quadratic-cubic nonlinear terms in the governing coupled PDEs of motion renders hardening/stiffening behavior on the dynamic responses of the beam when traversed by a moving mass. In addition, the obtained nonlinear results are compared with those from the linear analysis.     

Influence of pre-buckling displacements on the elastic critical loads of thin walled bars of open cross section

Nonlinear expressions for the strains occurring in thin walled bars of open cross section, when subjected to axial, flexural and torsional displacements, are incorporated in a general instability analysis based on the vanishing of the second variation of the total potential energy. It is shown that the influence of the pre-buckling displacements is automatically included in the analysis. A closed form solution for the lateral buckling of a simply supported beam subjected to uniform bending agrees exactly with a solution based on the governing differential equations. Solutions obtained using numerical methods are also presented. The significance of the second order axial strains induced by rotation about the shear centre, is investigated by considering the instability of an inverted T-beam subjected to uniform bending.     

旋转变截面扭梁的变形应力分析

本文研究了在剪切变形和转动的影响下,在轴向变形、弯曲变形和剪切变形的作用下的旋转变截面扭梁的单元刚度矩阵。假设扭转角、宽度和厚度沿着梁的长度方向都是线性变化的。给出了转速和轮毂半径对于梁的变形和应力的影响：转速增大时能增加轴线方向的变形和应力,但是却减小弯曲变形、剪切变形以及由它们产生的应力；而轮毂半径主要是影响梁的轴线方向的变形和应力,对弯曲变形和剪切变形的影响比较小。     

Model study and active control of a rotating flexible cantilever beam

For a dynamic system of a rotating flexible cantilever beam, the traditional model assumes the small deformation in structural dynamics where axial and transverse displacements at any point in the beam are uncoupled. This traditional hybrid coordinate model is referred as the zero-order approximation coupling model in this paper, which may result in divergence to the dynamic problem of a flexible cantilever beam with a high rotational speed. In this paper, a first-order approximation coupling model is presented to analyze the dynamics of rotating flexible beam system, which is based on the Hamilton theory and the finite element discretization method. The proposed model for the system considers the second-order coupling quantity of the axial displacement caused by the transverse displacement of the beam. The dynamic characteristics of the rotating beam system when using the zero-order approximation coupling model are compared with those when using the first-order approximation coupling models through numerical simulations. In addition, the applicability of the two dynamic models for control design are studied by using the classical optimal control method. Simulation and comparison studies show that, for the case without control for the system, there exists big difference between the result using the zero-order approximation coupling model and that using the first-order approximation coupling model even for the case of small angular velocity of the system. The larger is the angular velocity, the bigger is the difference. Vibration frequency of the beam by using the first-order approximation coupling model is higher than that by using the zero-order approximation coupling model. When the angular velocity of the system is close to or is larger than the fundamental frequency of the beam without rotation motion, the zero-order approximation coupling results in a wrong result, while the first-order approximation coupling model is valid. For the case with control for the system, the applicability of the zero-order approximation coupling model can be much broadened. The critical angular velocity of the system for validity of the zero-order approximation coupling model is much larger than that without control for the system. The first-order approximation coupling model is available not only for the case of small angular velocity but also for the case of large angular velocity of the system, and is applicable to the cases with or without control for the system.     

计及变形耦合的双连杆柔性机械臂动力学模型

根据连续介质的几何非线性变形原理,结合柔性多体系统建模理论的最新进展,在柔性梁的纵向、横向变形位移中均考虑了横向弯曲变形以及轴向伸缩变形的耦合作用,对一双连杆柔性机械臂系统,将其简化为柔性梁结构,利用有限元法和Lagrange方程,得出不同于传统零次近似方程,也不同于一次耦合动力学方程的新方程, 并可将此方程拓展到多杆机械臂系统。模型包含了二次耦合附加项,并且由于增加了变形耦合量,对柔性梁产生了“软化”作用,使得柔性机械臂模型不同于传统动力学模型。最后,计算仿真说明了这种差异,揭示了新模型的特点和有效性。     

升降横移式立体车库力学分析与结构参数优化

现场测试以及对车库运行过程的力学分析表明,在整个车库结构中,纵梁的扭转变形和载车板的弯曲变形对车库运行的平稳性影响最大。针对这一问题,用ANSYS有限元分析软件对纵梁及载车板进行了受力分析,得到了纵梁及载车板的载荷及变形分布,及其随起吊点位置的变化规律;给出了提高纵梁刚度的具体措施,有效地改善了车库的平稳性。     

Investigation of dynamic characteristics of a valve train system

A valve train dynamics model of internal combustion engine has been developed using the kineto-elastodynamics method. The dynamics behavior for flexible components such as the valve springs in the valve train system was described by the wave equation. The contact force at the cam/tappet interface was estimated by the elasto-hydrodynamic lubrication theory of finite line conjunction. Component submodels were integrated into the whole valve train model by coupling the corresponding contact and friction forces, and solved simultaneously considering transient effect of lubrication, as well as the torsional and bending vibrations of camshaft. Experimental validation was performed on a motored test rig. The predicted results agree well with experiments at difference camshaft speeds. The effects of the bending deflection and torsional vibration of camshaft on performance of the valve train system were also studied.     

A three-dimensional model for the dynamics of micro-endmills including bending, torsional and axial vibrations

Attainable geometric accuracy and surface finish in a micromilling operation depends on predicting and controlling the vibrations of micro-endmills. The specific multi-scale geometry of micro-endmills results in complexities in dynamic behavior, including three-dimensional vibrations, which cannot be accurately captured using one-dimensional (1D) beam models. This paper presents an analytically based three-dimensional (3D) model for micro-endmill dynamics, including actual cross-section and fluted (pretwisted) geometry. The 3D model includes not only bending, but also coupled axial/torsional vibrations. The numerical efficiency is enhanced by modeling the circular cross-sectioned shank and taper sections using 1D beam equations without compromising in model accuracy, while modeling the complex cross-sectioned and pretwisted fluted section using 3D linear elasticity equations. The boundary-value problem for both 1D and 3D models are derived using a variational approach, and the numerical solution for each section is obtained using the spectral-Tchebychev (ST) technique. Subsequently, component mode synthesis is used for joining the individual sections to obtain the dynamic model for the entire tool. The 3D model is validated through modal experimentation, by comparing natural frequencies and mode-shapes, for two-fluted and four-fluted micro-endmills with different geometries. The natural frequencies from the model was seen to be within 2% to those from the experiments for up to 90 kHz frequency. Comparison to numerically intensive, solid-element finite-elements models indicated that the 3D and FE models agree with less than 1% difference in natural frequencies. The 3D-ST model is then used to analyze the effect of geometric parameters on the dynamics of micro-endmills.     

Determination of the principal directions of composite helicopter rotor blades with arbitrary cross sections

Modern helicopter rotor blades with non-homogeneous cross sections, composed of anisotropic material, require highly sophisticated structural analysis because of various cross sectional geometry and material properties. They may be subjected by the combined axial, bending, and torsional loading, and the dynamic and static behaviors of rotor blades are seriously influenced by the structural coupling under rotating condition. To simplify the analysis procedure using one dimensional beam model, it is necessary to determine the principal coordinate of the rotor blade. In this study, a method for the determination of the principal coordinate including elastic and shear centers is presented, based upon continuum mechanics. The scheme is verified by comparing the results with confirmed experimental results.     

多级行星齿轮系统耦合动力学分析与试验研究

基于齿轮啮合理论和Lagrange方程,考虑各级齿圈扭转支撑刚度,提出运用集中参数法建立多级行星齿轮—箱体耦合扭转动力学模型。在分析多级行星齿轮啮合相位关系的基础上,确定各齿轮对时变啮合刚度,并运用有限元法获取各齿圈扭转支撑刚度。行星齿轮传动误差表示为轴频和齿频叠加的谐波函数,分析多级行星齿轮传动的主要激励特征。针对盾构机三级行星减速器某施工地段的运行条件,求解行星齿轮系统的动态响应,并分析其时频特性。采用背靠背能量回馈试验台架测试方案,测量盾构机行星减速器的振动加速度,采用数值积分计算振动速度和位移,并分析其振动特性。研究表明盾构机行星减速器振动试验数据与计算结果具有良好的一致性,验证多级行星齿轮系统耦合动力学模型的准确性。     

考虑温度效应的两端固支微机电开关梁静力分析

建立考虑温度效应的二维两端固支微机电开关梁静力变形分析模型,利用该模型分析温度变化引起的失稳问题,并利用该模型对两端固支微开关梁受电场力和温度作用下的受力变形进行分析。分析过程分为三个阶段：梁受电场力和温度变化作用,而在梁中点不受任何约束的变形阶段;除受电场力和温度变化作用外,梁中点变形受约束,因而梁中点受支反力作用的变形阶段;梁受电场力、温度变化以及支反力作用,并且在梁的中间一段具有指定挠度和转角的变形阶段。第二阶段中支反力的大小以及第三阶段中支反力的大小和位置都同驱动电压和温度变化有关。在实例中,给出部分计算结果,包括：温度变化对不同阶段梁受力变形的影响、对吸合电压的影响、对微开关电容的影响,以及不同温度变化下梁长对吸合电压的影响和不同温度变化下梁高对吸合电压的影响等,从中可以看出温度变化的作用及考虑温度效应的必要性。     

Theoretical Model for Friction-Induced Vibration of Ball Joint System under Mode-Coupling Instability

A theoretical model for the friction-induced vibration in ball joint systems is proposed. In a ball joint system, a small contact area is formed on the surface of the ball due to clearance between the ball and hemispherical cup. To analyze the dynamic instability induced by friction in the ball joint system, the dynamic beam-on-socket model with nonconformal contact under three-dimensional rotation was theoretically proposed. The results from stability analysis revealed that the mode-coupling type instability occurred for the lower bending modes at a specific combination of rotation components as well as an increase in the friction coefficient. In particular, the spinning component was essential in generating the mode-coupling instability between the doublet bending modes of the beam.     

On the lateral instability of a thin beam under periodic bending loads

The lateral instability of a thin beam under periodic bending loads was investigated. Physical evidences of the instability were observed previously by experiments. But an analytical study has not been reported. The object of this study is to demonstrate the nature and existence of dynamic lateral instability. The harmonic balance method is applied to bifurcation modes which result from the stability change of torsional mode of a beam and then compared with numerical simulations. It is found, in a certain frequency range, that a small bending load results in the lateral instability when damping is small. Inha University     

Coupled bending and torsional vibration of a rotor system with nonlinear friction

Unacceptable vibrations induced by the nonlinear friction in a rotor system seriously affect the health and reliability of the rotating machinery. To find out the basic excitation mechanism and characteristics of the vibrations, a coupled bending and torsional nonlinear dynamic model of rotor system with nonlinear friction is presented. The dynamic friction characteristic is described with a Stribeck curve, which generates nonlinear friction related to relative velocity. The motion equations of unbalance rotor system are established by the Lagrangian approach. Through numerical calculation, the coupled vibration characteristics of a rotor system under nonlinear friction are well investigated. The influence of main system parameters on the behaviors of the system is discussed. The bifurcation diagrams, waterfall plots, the times series, orbit trails, phase plane portraits and Poincaré maps are obtained to analyze dynamic characteristics of the rotor system and the results reveal multiform complex nonlinear dynamic responses of rotor system under rubbing. These analysis results of the present paper can effectively provide a theoretical reference for structural design of rotor systems and be used to diagnose selfexcited vibration faults in this kind of rotor systems. The present research could contribute to further understanding on the self-excited vibration and the bending and torsional coupling vibration of the rotor systems with Stribeck friction model.     

Active robust vibration control of flexible composite beams with parameter perturbations

In this paper, a time domain approach is presented to treat the problem of active controlling simultaneously the bending and torsional vibration of flexible composite beams under both mode truncations and parameter perturbations. In the proposed approach, the residual model, which is known as unstructured uncertainty, is viewed as an additive perturbation to the controlled model. Based on a state space model, which incorporates both bending and torsional deformation effects, of the flexible composite beam with piezoelectric sensors and actuators, a robust stability condition is derived to guarantee that both bending and torsional vibration of the flexible composite beam, which is subject to both mode truncation and linear time-varying parameter perturbations, can be actively controlled by an observer-based controller. The proposed robust stability condition gives an insight into the relationship between the stability margins of the controlled and residual mode subsystems, spillover effects and additive time-varying parameter perturbations. Finally, an active robust vibration control problem of a cantilevered flexible composite beam with piezoelectric sensors and actuators is provided for illustration.     

Flexural–torsional behavior of thin-walled composite space frames

A general analytical model based on the first-order shear deformable beam theory applicable to thin-walled composite space frames with arbitrary lay-ups under external loads is presented. This model accounts for all the structural coupling coming from the material anisotropy. The seven governing equations are derived from the principle of the stationary value of total potential energy. A displacement-based one-dimensional 14 degree-of-freedom space beam model which includes the effects of shear deformation, warping is developed to solve the problem. Numerical results are obtained to investigate the effects of fiber orientation on flexural–torsional responses of thin-walled composite space frame under vertical load.     

单根带发动机附件驱动系统模型及其求解

单根带发动机附件驱动系统广泛应用到汽车工业中。根据是否考虑带段横向振动,将SBAD系统动力模型归纳为旋转运动模型和旋转—横向运动耦合模型两种类型,其中耦合模型又根据是否考虑带段的弯曲刚度分为弦线耦合模型和梁耦合模型;分析、总结了动力模型的特点;接着回顾、总结了SBAD系统动力模型的求解技术;最后对SBAD系统研究中存在的不足及进一步研究的重点进行了探讨,指出为更真实地描述SBAD系统,建模时应考虑系统的各种阻尼和SBAD系统新技术,及引入张紧器迟滞模型。     

Damage detection of damaged beam by constrained displacement curvature

Structural damage detection technique involves the problem of how to locate and detect damage that occurs in a structure by using the observed changes of its dynamic and static characteristics. The objective of this study is to present an analytical method for damage detection by utilizing displacement curvature and all static deflection data to be expanded from the measured deflection data. Utilizing the measured displacements as displacement constraints to describe a damaged beam, minimizing the change of the displacement vector between undamaged and damaged states, and neglecting the variation between the test data and the analytical results, the deflection shape and displacement curvature of the damaged beam can be estimated, and the damage can be detected by the curvature. The validity and effectiveness of the proposed method are illustrated through comparison with the experimental results of a simple cantilever beam test.     

考虑摩擦作用的多柔体系统点--面碰撞模型

提出了考虑摩擦作用的多柔性体系统的点－面碰撞模型,基于Ｈｅｒｔｚ接触理论和非线性阻尼项描述法向碰撞接触过程的动力学特性,利用线性届接触刚度考虑碰撞过程中的摩擦作用。针对作大范围回转运动的柔性梁与一固定斜面发生碰撞的情况,在考虑刚柔耦合效应的多柔性系统动力学建模理论的基础上,精细研究了系统斜碰撞过程中的动力学行为。     

Hybrid dynamic modeling and analysis of a ball-screw-drive spindle system

We propose a comprehensive modeling approach to investigate the rigid-flexible coupling vibration of a Ball-screw-drive spindle system (BSDSS). Taking into account the rigid-flexible coupling between the ball screw and rigid components, a hybrid dynamic model for the BSDSS is established. The ball screw shaft was modeled by a flexible beam with axial, torsional, and transverse deformations, and the Ritz series expansion was used to approximate the continuous deformations. The motion equations were obtained using the Lagrange method. The analytical solution was derived and verified. According to the proposed model, the dynamic behavior of the system for various operating conditions was simulated and discussed. The numerical simulation result is useful for the controller design or tuning and the stiffness matching design of ball screw drives.