多约束状态下重载机械式主轴有限元建模及模态分析

重载机械式主轴是重型数控(Computer numerical control,CNC)机床摆角铣头的关键功能部件,具有大功率、大扭矩特点,用于加工大型复杂曲面零件。其有限元建模及模态分析的准确性是主轴进一步动力学分析的基础。基于Timoshenko梁理论建立重载机械式主轴的运动方程,采用有限元法得到主轴的矩阵形式的动力学方程;在有限元软件中分别以实体单元和梁单元对主轴进行有限元建模,对轴承以Combin14弹簧单元建模,并以自由模态和实际工况约束条件下进行重载机械式主轴的模态分析;根据轴承型号计算轴承的径向刚度,作为重载主轴模态分析中弹簧单元的刚度参数;进行主轴锤击模态试验,验证重载机械式主轴多约束状态下模态分析建模及仿真结果的正确性。研究表明,把主轴考虑为Timoshenko梁单元和Beam188梁单元进行主轴有限元建模和模态分析时,结果更为准确,弹簧约束梁方式更符合实际情况。研究结果为重载机械主轴系统的进一步优化设计和精度控制提供依据。     

Non-classical stiffness strengthening size effects for free vibration of a nonlocal nanostructure

New analytical solutions for free vibration of thick nanostructures are presented based on the nonlocal elastic stress field theory and the Timoshenko shear deformable nanobeam model. By applying the variational principle, new governing equations of motion and higher-order boundary conditions for these thick nanobeams are derived and their physical characteristics interpreted. The nonlinear history of straining involving higher-order strain gradients is considered in the derivation of strain energy and the contribution of higher-order strain gradients results in non-classical equations of motion thereby indicating that direct replacement of stress and moment quantities into the classical equations of motion is invalid. The Timoshenko nanobeam models are well suited for modeling and investigating the nonlocal behaviors of size-dependent carbon nanotubes. The effects of nanobeam size and various boundary conditions including simple supports, free and clamp constraints, such as a cantilevered nanotube, on the natural vibration frequency of nanotubes are discussed. The effects of nonlocal nanoscale are confirmed by comparing with molecular dynamic simulation solutions for (5,5) and (10,10) carbon nanotubes with four types of boundary conditions. The influence by nanoscale effect on the frequency ratio of nanotubes with different diameters is investigated. Further analysis based on the analytical nonlocal Timoshenko nanobeam model and the Euler–Bernoulli nanobeam model shows that the frequency ratio is more sensitive to nonlocal effect for free vibration of a nonlocal nanostructure if shear deformation is considered.     

A mechanically based approach to non-local beam theories

A mechanically based non-local beam theory is proposed. The key idea is that the equilibrium of each beam volume element is attained due to contact forces and long-range body forces exerted, respectively, by adjacent and non-adjacent volume elements. The contact forces result in the classical Cauchy stress tensor while the long-range forces are modeled as depending on the product of the interacting volume elements, their relative displacement and a material-dependent distance-decaying function. To derive the beam equilibrium equations and the pertinent mechanical boundary conditions, the total elastic potential energy functional is used based on the Timoshenko beam theory. In this manner, the mechanical boundary conditions are found coincident with the corresponding mechanical boundary conditions of classical elasticity theory. Numerical applications are also reported.     

Thermal post-buckling analysis of slender columns using the concept of coupled displacement field

Thermal post-buckling analysis of columns with an axially immovable ends is studied using the Rayleigh-Ritz (R-R) method, where the admissible displacement functions are chosen based on the concept of coupled displacement field (CDF) criteria. Geometric non-linearity is considered using the von-Karman strain displacement relations of the beam. Furthermore, the displacement fields derived from CDF criteria are used in an intuitive formulation, where the thermal post-buckling behavior can be predicted by using two parameters namely tension developed in the column and linear buckling load. An exhaustive set of column boundary conditions are considered namely classical such as hinged-hinged, clamped-clamped, clamped-hinged and non-classical such as clamped-guided and hinged-guided. Post-buckling analysis results are presented in the form of closed form expressions, where the ratio of post-buckling load to linear buckling load parameter is expressed as a function of central amplitude of the column for all the boundary conditions considered. The amount of non-linearity predicted using the present formulations (R-R method and intuitive method) based on the concept of coupled displacement field (CDF) criteria shows an excellent agreement with the available literature results for both classical and non-classical boundary conditions.     

Large amplitude free vibration analysis of Timoshenko beams using a relatively simple finite element formulation

Free vibration analysis of uniform isotropic Timoshenko beams with geometric nonlinearity is investigated through a relatively simple finite element formulation, applicable to homogenous cubic nonlinear temporal equation (homogenous Duffing equation). Geometric nonlinearity is considered using von-Karman strain displacement relations. The finite element formulation begins with the assumption of the simple harmonic motion and is subsequently corrected using the harmonic balance method. Empirical formulas for the non-linear to linear radian frequency ratios, for the boundary conditions considered, are presented using the least square fit from the solutions of the same obtained for various central amplitude ratios. Numerical results using the empirical formulas compare very well with the results available from the literature for the classical boundary conditions such as the hinged–hinged, clamped–clamped and clamped–hinged beams. Numerical results for the beams with non-classical boundary conditions such as the hinged-guided and clamped-guided, hitherto not studied, are also presented.     

Active disturbance rejection boundary control of Timoshenko beam with tip mass

In this paper, the active disturbance rejection control (ADRC) is utilized to stabilize the vibration of perturbed Timoshenko beam model with tip mass. The boundary control design is based on a hybrid PDE–ODE model, and is accompanied with designing a high-gain extended state observer (ESO) that is used to estimate the boundary disturbances. By transforming the model into the appropriate state space, the semigroup theory is employed to prove the well-posedness of the closed-loop system. To this end, it is proved by a frequency domain method that the semigroup generated by the system operator is exponentially stable, which allows to conclude the boundedness of perturbed closed-loop system response. The stability of the closed-loop system is further analyzed using the Lyapunov approach. Simulation results are presented to illustrate the efficacy of the suggested method.     

弹性地基上转动功能梯度材料Timoshenko梁自由振动的微分变换法求解

基于Timoshenko梁理论研究弹性地基上转动功能梯度材料（FGM）梁的自由振动。首先确定功能梯度材料Timoshenko梁的物理中面,利用广义Hamilton原理推导出该梁在弹性地基上转动时横向自由振动的两个控制微分方程。其次采用微分变换法（DTM）对控制微分方程及其边界条件进行变换,计算了弹性地基上转动功能梯度材料Timoshenko梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种不同边界条件下横向自由振动的量纲一固有频率,与已有文献的计算结果进行比较,退化后结果一致。最后讨论了不同边界条件、转速、弹性地基模量和梯度指数对功能梯度材料Timoshenko梁自振频率的影响。结果表明：功能梯度材料Timoshenko梁的量纲一固有频率随量纲一转速和量纲一弹性地基模量的增大而增大;在量纲一转速和量纲一弹性地基模量一定的情况下,梁的量纲一固有频率随着功能梯度材料梯度指数的增大而减小。     

Parametric instability of spinning twisted Timoshenko beams under compressive axial pulsating loads

The parametric instability on lateral bending vibrations of a spinning pretwisted beam under compressive axial pulsating forces is investigated. Equations of motion of the twisted beam are derived in the spinning twist coordinate frame using the Timoshenko beam theory and applying the Hamilton’s principle. The finite element method is employed to discretize the equations of motion into time-dependent ordinary differential equations with gyroscopic terms. A set of second-order ordinary differential equations with periodic coefficients of Mathieu-Hill type is formed to obtain the boundary frequencies of instability regimes. The influence of twist angle, spinning speed, static component of axial force, aspect ratio and restraint condition on the instability regions of the spinning twisted Timoshenko beam is discussed.     

几何非线性分析的二维共旋铁摩辛柯梁单元

改进了独立于单元的共旋(EICR)二维梁列式,使得不必再重新推导局部共旋标架下的单元材料刚阵,该列式可直接将现有性能良好的线性梁单元扩展用于二维梁结构任意大转动的几何非线性分析,同时推导了随动压强载荷作用引起的载荷刚度项,确保了高效的收敛速率。基于二维EICR梁列式将一种工程中实用的铁摩辛柯梁单元扩展用于几何非线性分析,数值算例表明文中所提共旋列式的铁摩辛柯梁单元计算精度高、计算效率高,可用于二维梁结构的几何非线性优化。     

Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform

This study investigates the vibration problems of an axially loaded non-uniform spinning twisted Timoshenko beam. First, using the Timoshenko beam theory and Hamilton's principle, we derive the governing equations and boundary conditions of the beam. Secondly, the differential transform method is used to solve these equations with appropriate boundary conditions. Finally, the effects of the twist angle, spinning speed, and axial force on the natural frequencies of a non-uniform Timoshenko beam are investigated and discussed.     

激光微加工系统中控制器的设计

在自行研制的激光微加工系统中,激光光源固定,待加工工件置于X-Y移动平台上,驱动平台运动带动工件运动,利用光束与工件的相对运动加工出各种图形.X-Y移动平台运动控制的核心是控制器.控制器通过串行通讯接口与PC机串行通讯,接收PC机下载的加工轨迹文件和运动控制命令,以步进电动机作为执行机构、以光栅尺作为反馈部件构成闭环控制系统.以光栅尺测量为标准,系统的位置控制误差小于2μm.     

C0-continuous isoparametric Timoshenko beam element for rotating

This paper presents a C0-continuous isoparametric finite element for free vibration analysis of a rotating, tapered Timoshenko beam mounted on the periphery of a rotating rigid hub, at a setting angle with the plane of rotation. The finite element has three nodes and two degrees of freedom per node and employs modified shape functions for rotational displacement associated with the shear strain energy to avoid shear locking. To obtain a finite element equation of the generalized eigenvalue problem, Hamilton’s principle is applied to kinetic and potential energy expressions of a rotating Timoshenko beam with non-zero setting angle. The numerical solutions for various situations including variations of rotational speed, taper ratio, slenderness ratio, hub radius and setting angle are compared with other numerical results available in the literature whenever possible. The results show that the new 3-noded isoparametric element yields accurate results when compared to other numerical ones.     

DYNAMIC BOUNDARY CONTROL OF BEAMS USING ACTIVE CONSTRAINED LAYER DAMPING

A globally stable boundary control strategy is developed to damp the vibration of beams fully treated with active constrained layer damping (ACLD) treatments. The devised boundary controller is compatible with the operating nature of the ACLD treatments where the strain induced generates a control force and moment acting at the boundary of the treated beam. The development of the boundary control strategy is based on a distributed-parameter model of the beam/ACLD system in order to avoid the classical spillover problems resulting from using ‘truncated’ finite element models. Such an approach makes the boundary controller capable of controlling all the modes of vibration of the ACLD-treated beams and guarantees that the total energy norm of the system is decreasing continuously with time. The control strategy is provided also with a dynamic compensator to shape the vibration damping characteristics of the ACLD in the frequency domain. The effectiveness of the ACLD in damping out the vibration of cantilevered beams is determined for different control gains and compared with the performance of conventional passive constrained layer damping (PCLD). The results obtained demonstrate the high damping characteristics of the boundary controller particularly over broad frequency bands.     

Flow-induced vibration and stability analysis of multi-wall carbon nanotubes

The free vibration and flow-induced flutter instability of cantilever multi-wall carbon nanotubes conveying fluid are investigated and the nanotubes are modeled as thin-walled beams. The non-classical effects of the transverse shear, rotary inertia, warping inhibition, and van der Waals forces between two walls are incorporated into the structural model. The governing equations and associated boundary conditions are derived using Hamilton’s principle. A numerical analysis is carried out by using the extended Galerkin method, which enables us to obtain more accurate solutions compared to the conventional Galerkin method. Cantilevered carbon nanotubes are damped with decaying amplitude for a flow velocity below a certain critical value. However, beyond this critical flow velocity, flutter instability may occur. The variations in the critical flow velocity with respect to both the radius ratio and length of the carbon nanotubes are investigated and pertinent conclusions are outlined. The differences in the vibration and instability characteristics between the Timoshenko beam theory and Euler beam theory are revealed. A comparative analysis of the natural frequencies and flutter characteristics of MWCNTs and SWCNTs is also performed.     

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.     

Natural frequencies,modes and critical speeds of axially moving Timoshenko beams with different boundary conditions

In this paper, natural frequencies, modes and critical speeds of axially moving beams on different supports are analyzed based on Timoshenko model. The governing differential equation of motion is derived from Newton's second law. The expressions for various boundary conditions are established based on the balance of forces. The complex mode approach is performed. The transverse vibration modes and the natural frequencies are investigated for the beams on different supports. The effects of some parameters, such as axially moving speed, the moment of inertia, and the shear deformation, are examined, respectively, as other parameters are fixed. Some numerical examples are presented to demonstrate the comparisons of natural frequencies for four beam models, namely, Timoshenko model, Rayleigh model, Shear model and Euler–Bernoulli model. Finally, the critical speeds for different boundary conditions are determined and numerically investigated.     

Modeling and bending vibration control of nonuniform thin-walled rotating beams incorporating adaptive capabilities

This paper addresses the problems of modeling and bending vibration control of tapered rotating blades modeled as nonuniform thin-walled beams and incorporating adaptive capabilities. The blade model incorporates non-classical features such as transverse shear, secondary warping and includes the centrifugal and Coriolis force fields. For the non-adaptive system, an assessment of a number of non-classical features including the taper characteristics is accomplished. The adaptive capabilities are provided by a system of piezoactuators bonded to the structure surface and spread over the entire span of the beam. Based on the converse piezoelectric effect and on the out-of-phase actuation, the piezoactuators produce a localized strain field in response to the applied voltage, and consequently, a change of the dynamic response characteristics is induced. A combined feedback control law relating the piezoelectrically induced transversal bending moment at the beam tip with the kinematical response quantities appropriately selected is used, and the beneficial effects upon the closed-loop dynamic characteristics of the blade are highlighted.     

微压入仪的设计与柔度分析

设计了基于仪器化压入技术的具有纳米测试精度的微压入仪,仪器集驱动、检测、控制和数据分析等功能于一体.采用音圈电机加载,无传动机构,减少了中间误差;通过高精度电容式位移传感器和应变式载荷传感器采集信号;设计了基于PID算法的闭环控制系统,编写了控制软件.通过有限元方法分析了加载时机架和试样的变形,标定了仪器柔度,仪器总体柔度约为1.8×10-8m/N,远小于目前商用仪器化压入仪的柔度,从原理上保证了测试结果的准确性.     

Instability and vibration of a rotating Timoshenko beam with precone

A rotating blade with a precone angle is usually designed, but little literature has investigated the effect of the precone angle on vibration. This paper investigates divergence instability and vibration of a rotating Timoshenko beam with precone and pitch angles. It uses Hamilton's principle to derive the coupled governing differential equations and boundary conditions for a rotating Timoshenko beam. Analytical solution of an inextensional Timoshenko beam without taking into account the Coriolis force effect can be derived. Some simple relations among the parameters of rotating Timoshenko beams are revealed. Based on these relations, one can predict the natural frequencies and parameters of other systems from those of known systems. Moreover, the mechanism of divergence instability (tension buckling) is investigated. Finally, the effects of the parameters on natural frequencies, and the phenomenon of divergence instability are investigated.     

Modeling and vibration feedback control of rotating tapered composite thin-walled blade

This paper addresses the problem of the modeling and vibration control of tapered rotating blade modeled as thin-walled beams and incorporating damping capabilities. The blade model incorporates non-classical features such as anisotropy, transverse shear, secondary warping and includes the centrifugal and Coriolis force fields. For the rotating blade system, a thorough validation and assessment of a number of non-classical features including the taper characteristics is accomplished. The damping capabilities are provided by a system of piezoactuators bonded or embedded into the structure and spread over the entire span of the beam. Based on the converse piezoelectric effect, the piezoactuators produce a localized strain field in response to a voltage and consequently, a change of the dynamic response characteristics is induced. A velocity feedback control law relating the piezoelectrically induced transversal bending moment at the beam tip with the appropriately selected kinematical response quantity is used and the beneficial effects upon the closed-loop dynamic characteristics of the blade are highlighted.