Non-linear free vibrations of Kelvin–Voigt visco-elastic beams

A full visco-elastic non-linear beam with cubic non-linearities is considered, and the governing equations of motion of the system for large amplitude vibrations are derived. By using the method of multiple scales, the non-linear mode shapes and natural frequencies of the beam are then analytically formulated. The resulting formulations for amplitude, non-linear natural frequencies and mode shapes can be used for any type of boundary conditions. Next, method of Galerkin is used to separate the time and space variables. The equations of motion show the presence of a non-linear damping term in addition to the ones with non-linear inertia and geometry. As it is known, the presence of non-linear inertia and the geometric terms make the non-linear natural frequencies to be dependent on constant amplitude of vibration. But, when damping non-linearities are present, it is seen that the amplitude is exponentially time-dependent, and so, the non-linear natural frequencies will be logarithmically time-dependent. Additionally, it is shown that the mode shapes will be dependent on the third power of time-dependent amplitude. The analytical results are applied to hinged–hinged and hinged–clamped boundary conditions and the results are compared with numerical simulations. The results match very closely for both cases specially for the case of hinged–hinged beam.     

Modeling of twist drills in terms of 3D angles

Traditional nomenclatures of specifying cutting tool geometries are two-dimensional (2D) in nature. The present work presents a paradigm to model the geometries of a variety of twist drills in terms of three-dimensional (3D) parameters. The work outlines the construction of a detailed computer-aided design (CAD) model for a fluted twist drill and establishes a new 3D definition for the geometry of drill in terms of biparametric surface patches. The flutes of the drill are modeled as helicoidal surfaces. For this, sectional geometry of tip-to-tip profile is developed and then swept. The geometric model of the shank is developed separately. The transitional surfaces are modeled as bicubic Bèzier surfaces. With this methodology, we propose a new 3D nomenclature for drill geometries in terms of 3D rotational angles. The relations necessary to map the proposed three-dimensional angles to two-dimensional conventional angles, known as forward mapping and their reverse relations (inverse mapping) are also developed. The new paradigm offers immense technological advantages in terms of numerous downstream applications.     

[BIM模型与智能设计]中低速磁浮系统起浮阶段的振动特性分析

为了研究电磁悬浮（EMS）中低速磁浮系统在起浮阶段的悬浮稳定性问题,建立了“车辆-电磁-轨道梁”耦合系统动力学模型。利用仿真模型分析了EMS中低速磁浮系统在起浮阶段的振动响应特性,对比分析了弹性梁与刚性梁的振动差别,仿真分析了单级悬浮控制与双级悬浮控制的效果差异。结果表明：EMS中低速磁浮列车在弹性轨道梁上起浮时,容易发生悬浮失稳的异常振动现象,原因是单级悬浮控制模式的悬浮架自激振动频率激发了弹性轨道梁的固有频率,从而诱发了车轨剧烈耦合共振;提高轨道梁刚度或采用双级悬浮控制器模式,可以缓解EMS中低速磁浮系统起浮阶段异常振动引起的悬浮失稳现象。     

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。     

Anwendung balkenförmiger schwingungstilger für biegeschwingungssysteme

Tuned viscoelastic dampers, comprising a mass connected through a viscoelastic link to a vibrating structure, have been considered from many points of view. For effective operation, the resonant frequency of such a damper must be made somewhat lower than the undamped resonant frequency of the structure. If the resonant frequency of the structure is low, some difficulty is encountered in constructing a good tuned damper system. In such a damper system the viscoelastic materials must be both soft and strong enough to give the necessary tuning frequencies without the damper being too heavy. For this reason a damper, based on a different principle for lateral vibrating structures is described in this paper. The main vibrating system is a clamped-free beam. The damper is also a clamped-free beam which is accomodated in an opening of the main system. The damper beam can vibrate freely in this opening and creates a new degree of freedom in the total system. The different equations for the lateral vibrations of the main and of the damper beam can be formulated according to the classical Bernoulli - Euler theory. For theoretical analysis, the system has been simplified. The main beam has partially constant rectangular sections. The auxiliary beam has a constant rectangular section. The total system has four different parts with constant sections. Structural damping of both beams has also been considered by a complex modulus of elasticity. For vibrations in a normal mode, the Euler - Bernoulli equation can be formulated as an ordinary differential equation. A general solution for each beam section is a sum of trigonometric and hyperbolic functions. The coefficients in the general expressions can be determined from the 16 boundary conditions of the beam damper system. The equations for the forced vibrations of the main beam, which can be excited by a prescribed motion at some point in the system or by a harmonic force are written in matrix notation:

$\text{|A| · |X| = |B|}$

The components of the matrix |A| are functions of the dimensionless frequency parameter and the geometric and the dynamic properties of the system. The matrix |B| contains the exciting functions, and the matrix |X| contains the unknown coefficients of the general solutions. For the solution of matrix |X| the Wilkinson method is used. The computed examples show, for a given beam-damper system, the amplitude of the vibration of the main beam as a function of frequency; the effect of tuning on the natural frequencies of the main beam; the effect of damping on the amplitudes of the main beam; the effect of the mass ratio MDamper/MBeam on the natural frequencies and resonant amplitudes of the main system; and the effect of the attaching point of the damper on the natural frequencies and resonant amplitudes of the main beam.     

Analytic solutions for vibration characteristics of a multi-beam structure

The first purpose of this paper is to mathematically analyze the vibration characteristics of four parallel and uniform beams that are joined by a tip body at their free ends. This beam structure is often found in many mechanical structures such as robots, space constructions, and optical pickup actuators in optical disc drives (ODDs). To analyze the vibration characteristics of this beam structure, this paper considers the motions in all six directions that are coupled to each other. Mathematical expansion using the energy conservation law applied to this beam structure yields four characteristic equations, which are related to a pure-axial (PA) vibration, a coupled bending-torsional (CBT) vibration, and two coupled axial-bending (CAB) vibrations. In addition, the correctness of the equations is verified by showing that the natural frequencies from these equations match those from a finite element (FE) analysis.The second purpose of this paper is to contribute to the performance improvement of an optical pickup actuator for ODDs by analyzing key parameters of the characteristic equations. This is one of the few papers that analytically derives solutions for the natural frequencies and modes of optical pickup actuators in all six directions. From the perspective of practical usefulness for the design of optical pickup actuators, this paper describes how to reduce rotation in the two CAB beam vibrations by investigating the relationship between design parameters and rotating properties.     

Natural frequencies of Euler-Bernoulli beam with open cracks on elastic foundations

A study of the natural vibrations of beam resting on elastic foundation with finite number of transverse open cracks is presented. Frequency equations are derived for beams with different end restraints. Euler-Bernoulli beam on Winkler foundation and Euler-Bernoulli beam on Pasternak foundation are investigated. The cracks are modeled by massless substitute spring. The effects of the crack location, size and its number and the foundation constants, on the natural frequencies of the beam, are investigated.     

MODELING APPROACHES AND SOFTWARE FOR PREDICTING THE PERFORMANCE OF MILLING OPERATIONS AT MAL-UBC

The author has been conducting research in the area of metal cutting mechanics, metal cutting dynamics, machine tool vibrations, precision machining and machine tool control in his Manufacturing Automation Laboratory, at The University of British Columbia, Canada since 1986. This article summarizes the research conducted in mechanics and dynamics of metal cutting in our laboratory. Modeling of mechanics of metal cutting is summarized first. The models include orthogonal to oblique cutting transformation, mechanistic modeling of cutting coefficients, slip line field and Finite Element modeling. The author mostly focused on milling. The kinematics of milling with and without structural vibrations is modeled. The geometric model of end mills and inserted cutters with arbitrary geometry are modeled. The prediction of forces, torque, power and dimensional surface finish is explained for milling operations. The chatter stability for milling operations is presented. The metal cutting knowledge is transferred to manufacturing industry by combining all the models in shop friendly software.     

Correct Prediction of the Vibration Behavior of a High Power Ultrasonic Transducer by FEM Simulation

This paper presents a study on three types of finite element analyses of high power ultrasonic transducer by using the finite element commercial software called ANSYS. The transducer geometry was treated as a 2D axi- symmetric model, 3D quart and full 3D model. For all of the simulations the modeled transducer was used in modal analysis and harmonic solutions to understand its mechanical behavior and its natural frequency. A comparison was made between each type of modeling and experimental results. This comparison allows the parameters of FEM models to be iteratively adjusted and optimized and also leads to selection of the best modeling type. The achieved FEM results exhibited a remarkably high predictive potential of ANSYS in modeling and simulation and allowed control on the design and on the vibration behavior of the high power ultrasonic transducer.     

Modeling and free vibration behavior of rotating composite thin-walled closed-section beams with SMA fibers

Smart structure with active materials embedded in a rotating composite thin-walled beam is a class of typical structure which is using in study of vibration control of helicopter blades and wind turbine blades. The dynamic behavior investigation of these structures has significance in theory and practice. However, so far dynamic study on the above-mentioned structures is limited only the rotating composite beams with piezoelectric actuation. The free vibration of the rotating composite thin-walled beams with shape memory alloy(SMA) fiber actuation is studied. SMA fiber actuators are embedded into the walls of the composite beam. The equations of motion are derived based on Hamilton’s principle and the asymptotically correct constitutive relation of single-cell cross-section accounting for SMA fiber actuation. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the Galerkin’s method. The formulation for free vibration analysis includes anisotropy, pitch and precone angle, centrifugal force and SMA actuation effect. Numerical results of natural frequency are obtained for two configuration composite beams. It is shown that natural frequencies of the composite thin-walled beam decrease as SMA fiber volume and initial strain increase and the decrease in natural frequency becomes more significant as SMA fiber volume increases. The actuation performance of SMA fibers is found to be closely related to the rotational speeds and ply-angle. In addition, the effect of the pitch angle appears to be more significant for the lower-bending mode ones. Finally, in all cases, the precone angle appears to have marginal effect on free vibration frequencies. The developed model can be capable of describing natural vibration behaviors of rotating composite thin-walled beam with active SMA fiber actuation. The present work extends the previous analysis done for modeling passive rotating composite thin-walled beam.     

Vibration analysis of viscoelastic single-walled carbon nanotubes resting on a viscoelastic foundation

Vibration responses were investigated for a viscoelastic Single-walled carbon nanotube (visco-SWCNT) resting on a viscoelastic foundation. Based on the nonlocal Euler-Bernoulli beam model, velocity-dependent external damping and Kelvin viscoelastic foundation model, the governing equations were derived. The Transfer function method (TFM) was then used to compute the natural frequencies for general boundary conditions and foundations. In particular, the exact analytical expressions of both complex natural frequencies and critical viscoelastic parameters were obtained for the Kelvin-Voigt visco-SWCNTs with full foundations and certain boundary conditions, and several physically intuitive special cases were discussed. Substantial nonlocal effects, the influence of geometric and physical parameters of the SWCNT and the viscoelastic foundation were observed for the natural frequencies of the supported SWCNTs. The study demonstrates the efficiency and robustness of the developed model for the vibration of the visco-SWCNT-viscoelastic foundation coupling system.     

Characterization and geometric modeling of single and overlapping craters in micro-EDM

In the micro-electro discharge machining (micro-EDM) process the machined surface consists of tiny overlapping craters. The shape of the crater has significant influence on the characteristics of the machined surface. However, existing models on crater shapes generally do not represent characteristic features of the craters, such as the bulging rim around the crater and the asymmetrical bowl-shape. In this study, experiments were conducted to examine three-dimensional (3D) geometrical features of craters in micro-EDM. Based on the experimental data, a geometric model of craters is proposed to better represent and understand the 3D shapes of single craters. A uniform model and a more realistic nonuniform model of crater geometry are developed, taking into account the bulging rim and the asymmetrical shape in craters. Furthermore, a method of constructing overlapping craters using nonuniform models of single craters is proposed and presented. This method is capable of representing major geometrical features of actual crater clusters.     

Postbuckling of 3D braided composite cylindrical shells under combined external pressure and axial compression in thermal environments

A postbuckling analysis is presented for a three-dimensional (3D) braided composite cylindrical shell of finite length subjected to combined loading of external pressure and axial compression in thermal environments. Based on a micro–macro-mechanical model, a 3D braided composite may be a cell system and the geometry of each cell is highly dependent on its position in the cross-section of the cylindrical shell. The material properties of epoxy are expressed as a linear function of temperature. The governing equations are based on a higher order shear deformation shell theory with a von Kármán–Donnell-type kinematic nonlinearity and includes thermal effects. A singular perturbation technique is employed to determine interactive buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, braided composite cylindrical shells with different values of shell geometric parameter and of fiber volume fraction under combined loading conditions. The results show that the shell has lower buckling loads and postbuckling paths when the temperature-dependent properties are taken into account. The effects of temperature rise, fiber volume fraction, shell geometric parameter, load-proportional parameter, as well as initial geometric imperfections are studied.     

具有局部表面损伤的滚动球轴承动力学建模

提出了一种滚动球轴承局部表面损伤故障的动力学分析方法。模型中每个轴承元件具有6个运动自由度,并完整地考虑了陀螺效应、离心力及润滑牵引等动力学因素。通过几何特征的变化对局部表面损伤进行了建模。采用四阶变步长Runge-Kutta-Fehlberg积分法对动力学方程进行了数值求解,并在时域和频域中对动力学响应进行了分析。研究结果表明,由于考虑了润滑剂的牵引效果,计算得到的故障特征频率较纯滚动假设下的计算结果有一定的差异。文中综合考虑了轴承元件的三维运动、相对滑动、润滑效应以及局部表面损伤等因素,因此所提出的模型更为全面、有效和实用。     

An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system

The aim of this study is to develop an approximate analytic solution for nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam with an attached heavy intra-span mass. A geometric nonlinearity due to midplane stretching is considered and Newton’s second law of motion along with Kelvin-Voigt rheological model, which is a two-parameter energy dissipation model, are employed to derive the nonlinear equations of motion. The method of multiple timescales is applied directly to the governing equations of motion, and nonlinear natural frequencies and vibration responses of the system are obtained analytically. Regarding the resonance case, the limit-cycle of the response is formulated analytically. A parametric study is conducted in order to highlight the influences of the system parameters. The main objective is to examine how the vibration response of a plain (i.e. without additional adornment) beam is modified by the presence of a heavy mass, attached somewhere along the beam length.     

基于传递矩阵法的连杆机构弹性动力学方程

本文从拉格朗日方程出发，在用传递矩阵法求解机构固有频率和主振型的基础上导出了连杆机构弹性动力学方程。与有限元法建立的方程相比，本方程模型精确、概念简明，求解便捷。     

Serial sectioning and digitization of porous media for two- and three-dimensional analysis and reconstruction

A procedure is proposed for serial sectioning of impregnated porous media, for photographing the sections under the microscope, and for digitizing the image using an IBM PC (XT) with a digitizing board. The result is a three-dimensional matrix containing the enhanced digitized image in a binary form of O's and I's which represent solid and pore spaces respectively. The procedure was tested on three different porous materials: (1) a pack of glass beads, 170–350 μm; (2) Berca sandstone; and (3) Elora clay loam soil. Each of these was impregnated, forty-six to eighty cross-sections were prepared by close parallel polishing of 10–50 μm per section, and then the sections were photographed and digitized. Examples of the real and digitized cross-sectioned samples are given as well as an example of reconstructed images in a plane perpendicular to the plane of sampling. It is proposed that the method of serial sectioning and automated digitization of porous media provides a powerful tool for further three-dimensional geometrical and topological investigation of pore space to be used in models of fluid transport phenomena.     

Dynamic analysis of three-dimensional helical geared rotor system with geometric eccentricity

A dynamic model of a multi-shaft helical geared rotor system is presented. Rotating shafts of the system are modeled as Timoshenko beams. A general three-dimensional dynamic model of helical gear pairs with geometric eccentricity is developed for the gear mesh and bearing flexibility is included in the model as well. The transmission error and gear geometric eccentricity are simulated as excitations. Eigenvalue solution and the modal summation technique are used to predict the natural frequencies and forced responses of the system. Then two geared rotor system models are presented for validation of the gear dynamic model. It is demonstrated that the gear mesh model is effective for general geared rotor systems, spur and helical gears, one-stage and multi-stage systems. Finally, forced responses of an example system are analyzed to demonstrate the influences of the helical gear geometric eccentricity and the coupling between gear geometric eccentricity and rotor mass unbalance.     

Investigations of complex vibrations of beams within the framework of the Sheremet’ev-Pelekh kinematic model using the wavelet transform

A model of the Sheremet’ev-Pelekh geometrically nonlinear beam is constructed with the help of the variational principle based on the method of hypotheses. The solutions to the derived sets of differential equations are analyzed on the basis of the qualitative theory of differential equations using wavelet analysis. A substantial difference in the nature of vibrations obtained using the Sheremet’ev-Pelekh kinematic model from the results of the Bernoulli-Euler and S.P. Timoshenko models is revealed.     

Semi-analytic methods for rotating Timoshenko beams

The equations of motion including shear and rotatory inertia are developed for uncoupled lead-lag and flapping vibrations of beams rotating at constant angular velocity in a fixed plane. Separability assumptions lead to an ordinary differential equation in the space variable, and a solution is obtained in terms of four independent functions, each a convergent power series. These beam functions are similar to classic normal beam functions, and application of the boundary conditions yields determinants whose roots are the natural frequencies. The simplicity and speed of this method is demonstrated by application to helicopter main rotor blades, and spoke diagrams and mass balancing are illustrated.