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.

     

A Study of the Effect of Normal Stiffness on Kinetic Friction Forces Between Two Bodies in Sliding Contact

This work determined what effect stiffness in the direction normal to the plane of contact between surfaces in sliding contact had on the phenomenon of stick-slip.

The results showed that the amplitude if stick-slip vibrations could be reduced by increasing the normal stiffness with contacting surfaces of mild steel.

It was also observed that with the above mentioned surfaces some wear had to take place before stick-slip would occur. This was attributed to the initial presence of oxide on the surfaces.     

Vibration monitoring in sliding wear of plasma sprayed ceramics

Dry frictional contact between two surfaces, one made of plasma sprayed ceramic coatings of Al₂O₃ and Al₂O₃---TiO₂ combination and the other made of steel, is analyzed. The experiments were conducted using a pin-on-disc set-up in the load range of 5–35 N and for sliding distances up to 14 km. The interactions between friction, wear and vibrations due to influence of normal load, sliding speed and system dynamics are investigated in the present paper. Two vibration parameters of pin in the load direction (vertical) are monitored, namely the r.m.s. acceleration and the kurtosis, which seem to be influenced considerably by the wear process and indicate correlation with wear mechanisms taking place such as stick-slip and grain pull-out, as evidenced by scanning electron microscopy of worn surfaces. The study shows that a range of frequency is to be utilized for vibration monitoring to include natural frequencis of the system consisting of pin in contact with disc. This could be estimated by a standard impulse hammer test. The pin acceleration decreases with increase in load and sliding distance, but with respect to sliding speed, the vibration level intially decreases but increases beyond the sliding speed of 1.5 m s⁻¹. Among the three ceramic coatings, TiO₂ is found to be most wear resistant, exhibiting the lowest friction coefficient and a low vibration level. Variation in kurtosis with run-in wear indicates smoothing of Al₂O₃ due to grain pull out.     

Experimental study on vibration characteristics of double-helical gearbox with isolators

A power closed double-helical gear test rig with isolators was created to investigate the dynamic response characteristics of the double-helical gearbox. Experimentally, the vibration accelerations of the bearing seat, machine foot, and base were tested under various working conditions. Experiments validated the validity of the generalized finite element theory-based dynamic model of the double-helical gear system. The study found that the unloaded vibration of the gearbox is greatly caused by the shaft frequency excitation due to assembly/manufacturing errors. As the load increases, the assembly/manufacturing errors are compensated by the system deformation. In the acceleration spectrum of the bearing seat, it is found that the tooth surface modification can reduce the amplitude of the mesh frequency, but the bearing outer ring frequency components are increased. That is, the axial movement of double-helical gears is intensified. Furthermore, the closer the excitation frequency is to the natural frequency of the base, the worse the vibration isolation effect is.

     

Stick-slip motion and surface deformation

The relation between stick-slip motion and surface deformation in the frictional contact of polymer films was studied by multiple-beam interferometry. The stick-slip motion is caused by the combined deformation behavior of semi-macroscopic contact regions within the apparent contact area.     

Tool wear rate prediction in ultrasonic vibration-assisted milling

Abstract

In the current study, a predictive model on tool flank wear rate during ultrasonic vibration-assisted milling is proposed. One benefit of ultrasonic vibration is the frequent separation between tool and workpiece as the cutting time is reduced. In order to account for this effect, three types of tool–workpiece separation criteria are checked based on the tool center instantaneous position and velocity. Type I criterion examines the instantaneous velocity of tool center under feed movement and vibration. If the tool is moving away from workpiece, there is no contact. Type II criterion examines the position of tool center. If the tool center is far from the uncut workpiece surface, there is no contact even though the tool is getting closer. Type III criterion describes the smaller chip size due to the overlaps between current and previous tool paths as a result of vibration. If any criterion is satisfied, the tool flank wear rate is zero. Otherwise, the flank wear rate is predicted considering abrasion, adhesion and diffusion. The proposed predictive tool flank wear rate model is validated through comparison to experimental measurements on SKD 61 steel with uncoated carbide tool. The proposed predictive model is able to match the measured tool flank wear rate with high accuracy of 10.9% average percentage error. In addition, based on the sensitivity analysis, smaller axial depth of milling, larger feed per tooth or higher cutting speed will result in higher flank wear rate. And the effect of vibration parameters is less significant.     

Active H∞ control of the vibration of an axially moving cantilever beam by magnetic force

An H_∞ method for the vibration control of an iron cantilever beam with axial velocity using the noncontact force by permanent magnets is proposed in the paper. The transverse vibration equation of the axially moving cantilever beam with a tip mass is derived by D'Alembert's principle and then updated by experiments. An experimental platform and a magnetic control system are introduced. The properties of the force between the magnet and the beam have been determined by theoretic analysis and tests. The H_∞ control strategy for the suppression of the beam transverse vibration by initial deformation excitations is put forward. The control method can be used for the beam with constant length or varying length. Numerical simulation and actual experiments are implemented. The results show that the control method is effective and the simulations fit well with the experiments.     

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。     

Optimal design of the gerotor (2-ellipses) for reducing maximum contact stress

The oil pump, which is used as lubricator of engines and auto transmission, supplies working oil to the rotating elements to prevent wear. The gerotor pump is used widely in the automobile industry. When wear occurs due to contact between an inner rotor and an outer rotor, the efficiency of the gerotor pump decreases rapidly, and elastic deformation from the contacts also causes vibration and noise. This paper reports the optimal design of a gerotor with a 2-ellipses combined lobe shape that reduces the maximum contact stress. An automatic program was developed to calculate Hertzian contact stress of the gerotor using the Matlab and the effect of the design parameter on the maximum contact stress was analyzed. In addition, the method of theoretical analysis for obtaining the contact stress was verified by performing the fluid-structural coupled analysis using the commercial software, Ansys, considering both the driving force of the inner rotor and the fluid pressure, which is generated by working oil.

     

Finite element modeling for stick-slip pattern of squeal modes in disc brake

This paper provides the finite element modeling for describing the nonlinear stick-slip response of squealing modes in a disc brake system. The analytical nonlinear contact kinematics is applied to each contact node of the finite element disc and pads. Numerical results show that a portion of the contact area can undergo the stick-slip oscillation. Depending on the size of the stick-slip zone, the corresponding squeal vibration can be either the pure harmonic or periodic oscillation. Particularly, the squeal mode arising from the pad rigid mode generates the periodic stick-slip limit cycle in its steady-squealing response, as opposed that the disc squeal modes become the pure harmonic response.     

Deformation and stability of an elastica under a point force and constrained by a flat surface

This paper studies the deformation and stability of a pinned elastica under a point force moving quasi-statically from one end to the other. The elastica is constrained by a rigid plane wall containing the two ends. Three types of equilibrium configurations can be found; they are non-contact, one-point contact, and one-line contact on the side. A vibration method is adopted to determine the stability of the calculated deformations. In order to take into account the variation of the contact region between the elastica and the plane wall during vibration, an Eulerian version of the governing equations is adopted. It is found that all the point-contact deformations are unstable. On the other hand, there are two different mechanisms a line-contact deformation becomes unstable; one through a secondary buckling and the other through a limit-point bifurcation. In the secondary buckling, the length of the line-contact segment and the axial force satisfy the Euler buckling criteria for a pinned-clamped column. On the other hand, when a line-contact deformation becomes unstable via a limit-point bifurcation, the axial force does not exceed the Euler buckling load. The theoretical predictions are confirmed by experimental observations.     

Flexural–torsional coupled vibration and buckling of thin-walled open section composite beams using shear-deformable beam theory

A general analytical model based on shear-deformable beam theory has been developed to study the flexural–torsional coupled vibration and buckling of thin-walled open section composite beams with arbitrary lay-ups. This model accounts for all the structural coupling coming from the material anisotropy. The seven governing differential equations for coupled flexural–torsional–shearing vibration are derived from Hamilton's principle. The resulting coupling is referred to as sixfold coupled vibration. Numerical results are obtained to investigate effects of shear deformation, fiber orientation and axial force on the natural frequencies, corresponding mode shapes as well as load–frequency interaction curves.     

Effect of blade tip pattern on blade load and vibration characteristics of a twin-stage axial flow fan

Focusing on a twin-stage axial fan, this paper investigates the effect of blade tip pattern on blade load and vibration characteristics. Steady simulations are first conducted to quantify the aerodynamic performance of various blade tip patterns. The finite element modeling analysis is performed to capture blade load and vibration characteristics, and Campbell diagram is introduced to evaluate resonance margin of different blade tip patterns. Results show that for all selected patterns, the first three mode shapes are mainly the bending of blade tip, which results in stress concentration at the blade root, while the last three are the waving in small range. The proposed blade tip patterns not only increase maximum stress and average deformation, but also significantly increase resonance margin near the rated speed. In addition, based on the harmonic response analysis, we find that the stress and amplitude frequency response will be notably altered by blade tip patterns.

     

Vibration suppression of atomic-force microscopy cantilevers covered by a piezoelectric layer with tensile force

In this paper, vibration suppression of a micro-beam covered by a piezoelectric layer is studied. The micro-beam is modeled with the specific attention to its application in AFM. The AFM micro-beam is a cantilever one which is stimulated close to its natural frequency by applying a harmonic voltage to the piezoelectric layer. The beam is an Euler-Bernoulli beam which abbeys Kelvin-Voigt model. Using such model supplies the comparison between elastic and viscoelastic beams; and one of the most important properties of viscoelastic materials, damping effect can readily be investigated. The pump provides an axial load with the result that it suppresses the vibrations. First, the vibration equations are extracted using Lagrangian and extended Hamiltonian method in vertical, longitudinal, as well as torsional directions and are discretized by exploiting the Galerkin mode summation approach. The discretized time-domain equations are solved by the aid of the Runge-Kutta method. The viscoelastic beam is compared with the elastic one, and the effects of damping ratio on vibration responses are presented. Additionally, the effects of micro-pump load, excitation voltage, and initial twist angle are investigated on the amplitude of vibration and natural frequency of system. It is observed that viscoelasticity of beam and axial load of the pump reduce vibrations and provide uniform time-domain responses without beatings.     

基于ABAQUS的滚珠丝杠传动刚度的计算分析

滚珠丝杠传动刚度是影响机床精度的关键因素之一,而丝杠的传动刚度,主要取决于该丝杠副在载荷的作用下丝杠的轴向变形、滚珠与滚道间的接触变形和支承轴承的轴向接触变形三者的大小.通过建立滚珠丝杠传动中关键零件受力分析的有限元模型,对滚珠丝杠传动三种主要轴向变形进行计算,并分析三种变形对丝杠传动刚度的影响大小,为提高丝杠的传动精度,改进滚珠丝杠副的性能提供了参考.     

Study on slip displacement and stick-slip characteristics in reciprocating friction drive system

The effects of follower mass, time ratio, driver speed, and normal load on the slip displacement and stick-slip characteristics are experimentally and theoretically investigated in the reciprocating friction drive system under dry contact using a 0.45% carbon steel pair. Results show that the accumulative slip displacement is linearly proportional to the rotating cycle under various operating conditions. The slope for this linear relationship is defined as the slip rate. The slip rate increases with increasing driver speed, the mass of the follower, and |δ−1|, but decreasing normal load. There are three modes of relative motion between the driver ring and the follower in the present study, namely, unstable stick-slip (USS), stable stick-slip (SSS), and sticking (ST) regimes. They are significantly influenced by the driver speed, normal load, time ratio and mass of the follower. The critical operating conditions among unstable stick-slip, stable stick-slip, and sticking regimes were also established. The critical frequency can be theoretically calculated and agrees well with experimental results.     

指尖密封弹塑性变形及刚度特性分析

为探讨指尖密封迟滞问题中是否存在塑性变形的影响,对指尖密封弹塑性变形进行了研究,通过有限元分析获得了指尖密封刚度随指尖密封结构参数变化的规律,并拟合出描述这一规律的数学公式。结果表明,指尖密封工作中存在的迟滞问题是由于作用在密封装置上下游流体压力差导致的指尖片和后挡板之间的摩擦阻滞作用而产生的,在指尖密封的结构参数和工况参数范围内,指尖曲梁并未因转子的径向碰撞而产生塑性变形,因此,指尖密封迟滞特性分析可以不考虑指尖曲梁是否发生有塑性变形;指尖曲梁的刚度随指尖曲梁轴向尺寸的增加和指尖数目的减少而增大。     

Chatter dynamics in sheet rolling

Two and four degree of freedom (DOF) systems describing chatter in sheet rolling are investigated. In the two DOF case, this includes studying the nonlinear behavior that arises due to the deformation at the work-backup roll interface. The force displacement relation between two cylindrical rolls being pressed together is inherently nonlinear due to the changing contact area. The investigation also includes the additional plastic deformation of the sheet in the roll-bite that is present during chatter. As the work rolls vibrate the gap widens and narrows inducing additional plastic deformation of the sheet.For the two DOF case, one mode of vibration is a motion of the work roll-sheet mass center. The other mode is a squeezing motion of the sheet in the bite by the work rolls. The natural frequencies seem to correspond with fifth octave chatter. The nonlinear theory predicts a small shift in the vibrational frequency. The role of inter-stand tension in triggering instability is also discussed.The four DOF system extends the research of Yarita et al. by providing analytic expressions for the natural frequencies. It appears that this system is capable of predicting third octave chatter, in addition to fifth octave chatter.     

Collision between a ring and a beam

With car–parapet collision accidents in mind, a normal collision between a free-flying half ring and a simply supported beam with/without axial constraints is studied, in which an elastic–plastic half ring with an attached mass and the elastic–plastic beam are taken as the simplest models of a car and a parapet, respectively. Particular attention is paid to the energy partitioning between the two structures and the evolution of the contact regions during collision. A mass–spring finite difference (MS–FD) model is employed whilst the large deflection and axial stretching/compression are incorporated. The numerical results show that the less stiff (i.e. softer) structure will dissipate more energy and the contact regions will move away from the initial contact points. With the increase of the relative thickness of the beam to the ring, the final deformation of the half ring will transform from a “U” shape to a “W” shape.     

The effect of variable amplitude loading on stress distribution within a cylindrical contact subjected to fretting fatigue

A finite element model of a cylindrical Hertzian contact on a test sample subjected to alternating shear loading has been developed. The model has been used to investigate shear stress distributions at the contact during variable amplitude fretting fatigue for a load configuration in which the sample cyclic stress is applied in phase with shear force on the cylindrical contact. It has been found that during constant amplitude cyclic loading, shear stress distributions and positions of the stick-slip boundary at load maxima and minima remain fixed. Application of overloads changes the stress distribution and the position of the stick-slip boundary attained by loading of subsequent cycles. The largest cycle maximum stress determines the position of the stick-slip boundary adopted by subsequent smaller amplitude cycles. In general variable amplitude fretting fatigue the position of the stick-slip boundary will be changing with each load cycle. Hence fatigue initiation processes will occur at locations dispersed over an extended region over the contact. The implications of this behaviour for models for fretting fatigue life calculation are explored.