Parametric vibrations of long cylindrical shells subject to transverse static load

The problem of parametric vibrations of long circular cylindrical shells prestressed by a uniformly distributed transverse static load has been considered. The static equilibrium position is found as a result of the solution of a nonlinear problem, where the factors contributing to nonlinearity include flattening of circular cross sections during deformation, stretching of elastic axis and hardening elastic foundation. Small amplitude forced parametric vibrations about the static position are then studied. The solution of the nonlinear parametric vibration problem in the absence of the static load is also shown.     

Parametric studies on buckling loads and critical speeds of microdrill bits

Transverse bending vibrations of the spinning microdrill bit subjected to a compressive axial load are developed based on the Timoshenko beam theory. The system equations of motion are discretized into the form of time-dependent ordinary differential equations by the finite element method. Two types of eigenvalue problems are formulated and utilized to study the effects of the drill helix angle, flute length and diameter on the buckling load and critical speed of microdrill bits with different supported ends. Equivalent formulae similar to those of untwisted Euler beams are established to predict critical buckling loads and critical speeds for microdrills and provide results with sufficient accuracy. The effect of rotational speed on the buckling load, and the influence of thrust force on critical speed are also investigated. A Galef-type equation associated with critical speed, thrust force and buckling load is formulated.     

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。     

Vibrations of belts and belt drives

During its operation, a power transmission belt can undergo axial, transverse and torsional motion. Models available for analyzing the free and forced vibrations of the belt are discussed. The effects of initial tension, transport velocity, bending rigidity, support flexibility, large displacements and belt and pulley imperfections are discussed. Vibrations of belt drives are also considered, and the influence of several complicating factors are mentioned. This article reviews the state-of-the-art in vibration analysis of power transmission belts. Many contributions found in the literature are brought together, and some new results are also included.     

Reinforcement minimization of beams with axial forces

Within the confines of absolute minimum weight design of plastic structures, it has generally been assumed that the relationship between material (cost) and the vector of internal actions is linear (or piecewise linear) that is, sandwich approximations, or essentially constant lever-arm assumptions for reinforced materials, have been employed. Where both axial load and moment act on the one cross-section, or moments are relatively large, such idealization of cost function is no longer consistent with actual behaviour. A nonlinear cost function model must be substituted. The cost-function for an idealized reinforced section under bending moment and axial load (tensile or compressive) is derived herein. Two examples of encastre reinforced beams with axial forces are considered. When the applied load consists of a transverse central point load the optimum design is found to accord with earlier results employing a simplified cost function. When the applied lateral load is uniformly distributed earlier results are only approached in the limit as the lateral load approaches infinity.     

圆螺纹套管接头的抗压缩极限载荷分析

采用非线性有限元方法，以API8牙圆螺纹套管接头为研究对象，在标准锥度公差条件下，模拟套管接头的上扣过程，计算分析上扣过程中内、外螺纹导向面和承载面上的应力分布，并对套管接头在压缩工况下的承载极限进行分析，得到套管接头在压缩载荷下的失效形式，失效载荷，并提出相应的失效判据。     

An analysis of the can effect in an isostatic pressing of copper powder

Based on an elastic-plastic analysis of the can deformation, experimental data from Wadley et al. (1991, Acta Metall. Mater.39, 979) are used to evaluate the shielding effect of the can in an isostatic pressing of copper powder. The calculation indicates that the consolidation of the powder compact is preceded by a stage of powder rearrangement. Thereafter, during consolidation, it is found that in this particular case, compressive stresses in the powder are about 20 MPa less than the pressure applied to the exterior of the can. Furthermore, the powder is not in a state of hydrostatic pressure. The axial compressive stress in the powder is less than the transverse compressive stress. This effect is due to the can which supports a larger load in the axial direction compared to the radial direction during plastic deformation. A model for the nonisostatic yielding of metal powder is compared to the data. It is found that strain hardening must be allowed for to get agreement between the model and the experiment for cold pressing.     

An Analysis of Flexible Seal Ring Vibrations

Vibration of the seal ring induced by the axial pulsation, wobbling and initial distortion of the rotor surfaces in a face-seal assembly consisting of a seal ring and a carrier are analyzed. The analyses are divided into two parts: a rigid body analysis dealing with vibrations due to the axial pulsation and wobbling; and an elastic ring analysis dealing with vibration induced by the initial surface waviness of the rotor. The fluid film characteristics are approximated by linear springs having two direct and two cross-coupling spring stiffnesses in the transverse and angular directions with negligible damping. Results are obtained in terms of natural frequencies and influence coefficients characterizing the response of the seal ring to the rotor motion.     

Experimental validation of appropriate axial immersions for helical mills

The forced periodic vibrations of milling processes with helical tools are modelled and analysed. By means of analytical and numerical techniques, the so-called superchart of milling is constructed that includes the conventional stability chart with the stability lobes and also the amplitudes of the stable forced vibrations, which is correlated to many surface quality numbers like surface location error, surface waviness and surface roughness. The existences of trivial and non-trivial appropriate axial immersions are presented for a single degree of freedom mechanical model of the machine tool structure. At these appropriate axial immersion parameters, if the cutting is stable, the surface quality parameters will be also optimal even in those spindle speed domains where the system is near to resonance and the cutting efficiency is high. Experiments are performed with a large industrial milling machine using a flexible test rig with an essential flexibility in one direction. The calculated and the measured forced vibration signals in the stable domains of the stability chart are in good correlation, which validates the simple analytical predictions for the non-trivial appropriate axial immersion parameters.     

Stability and dynamics of rotor system with 45° slant crack on shaft

Crack on a shaft is one of the common damages in a rotor system. In this paper, transverse vibrations are calculated to compare the influences of transverse crack and slant crack on the rotor system. Results show that the vibration amplitude of the rotor system with a 45° slant crack on the shaft is larger than that with a transverse crack when the two types of crack have the same depth and the rotor system runs in the same condition. Stability and dynamic characteristics of the rotor system with a 45° slant crack on the shaft under torsional excitation are analyzed by considering opening and closing of the crack. It is shown that the instability of the transverse vibration of the rotor system increases with increasing difference between the bending stiffness in two main directions, and the vibration is stable when the two bending stiffness are identical. The spectrum analysis of the steady-state response reveals that the gravity and the eccentricity produce different frequency components, and when the two bending stiffness are identical, the multiple frequency components of the torsional excitation disappear. Further investigation shows that the vibration amplitudes in combined frequencies increase rapidly in transversal, torsional, and axial vibration with increasing slant crack depth. The results are helpful for the understanding the dynamic behavior of a rotor system with a slant crack on a shaft and can be used for the detection of the slant crack on a shaft.     

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.     

A note on unimodal and bimodal optimal design of vibrating compressed columns

The bimodal optimization with respect to the frequency of transverse vibrations under axial compression is demonstrated on the simple Pragers' model. This model consists of rigid segments that are joined by elastic hinges to each other and to the supports and point masses which are situated in the inner hinges.     

Nonlinear bending of Reissner–Mindlin plates with free edges under transverse and in-plane loads and resting on elastic foundations

A nonlinear bending analysis is presented for a rectangular Reissner–Mindlin plate with free edges subjected to combined transverse partially distributed load and compressive edge loading and resting on a two-parameter (Pasternak-type) elastic foundation. The formulations are based on the Reissner–Mindlin plate theory considering the first-order shear deformation effect, and including the plate-foundation interaction. The analysis uses a mixed Galerkin-perturbation technique to determine the load–deflection curves and load–bending moment curves. Numerical examples are presented that relate to the performances of moderately thick rectangular plates with free edges subjected to combined loading and resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influence played by a number of effects, among them foundation stiffness, transverse shear deformation, loaded area, the plate aspect ratio and initial compressive load are studied. Typical results are presented in dimensionless graphical form.     

The dynamics of symmetric post-buckling

This work examines the relationship between natural frequency and compressive load for a conservative elastic mechanical system characterised by symmetric buckling. Application of some general theory shows that for a perfect system, exhibiting a stable-symmetric point of bifurcation, the initial post-buckling curve of compressive load against the square of the natural frequency is linear and has a slope of that of the pre-buckling curve. Simple link models and discretised continuous systems are analysed to illustrate the results of the general theory. These models are also used to show that the initial post-buckling curve is independent of axial inertia. The influence of initial imperfections is then considered and compared with experimental evidence.     

The Performance of Externally Pressurized Bearings Using Simple Orifice Restrictors

The load capacity and vibration characteristics of externally pressurized thrust and journal bearings have been investigated. The bearings used orifices sufficiently recessed back from the bearing surface to ensure that the area of the orifice is the minimum presented to the gas flow. The journal bearings consist of plain cylinders with one or two rows of eight orifices, and the thrust plates of plain discs with six orifices drilled in equispaced circumferential grooves.

The load capacity of the thrust plates agrees with that calculated but the vibrations are greater than predicted. The load capacity of journal bearings is about half that calculated assuming axial flow conditions in the bearing, but circumferential flow can account for the difference. Synchronous and half-speed vibrations of a shaft rotating within these bearings have been measured and found to agree with those calculated on the assumption that the gas films behave as linear springs.     

Transverse shear and rotary inertia effects on the stability analysis of functionally graded shells under combined static and periodic axial loadings

The effect of transverse shear and rotary inertias on the dynamic stability of functionally graded cylindrical shells subjected to combined static and periodic axial forces is investigated in this paper. Material properties of functionally graded cylindrical shells are considered temperature-dependent and are graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. Numerical results for silicon nitride-nickel cylindrical shells are presented based on two different methods: the first-order shear deformation theory (FSDT) which considers the transverse shear strains and the rotary inertias, and the classical shell theory (CST). The results obtained show that the effect of transverse shear and rotary inertias on the dynamic stability of functionally graded cylindrical shells subjected to combined static and periodic axial forces is dependent on the shell’s material composition, environmental temperature, amplitude of static load, deformation mode, and the shell’s geometry parameters.     

The elastically supported Timoshenko beam subjected to an axial force and a moving load

A solution for the flexural vibration of an elastically supported Timoshenko beam which is subjected to an axial force and a moving transverse load is obtained. The influences of the axial force and the load velocity on the beam response are studied and the characteristics of the various resonances are examined. The results are also compared with those by the Euler beam theory.     

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.     

Application d'un modèle de flambement conditionné à la pose des câbles à fibres optiques

Application of a conditioned buckling solution to the laying of optical cables. Mechanical behaviour of cables submitted to an axial compressive load is subjected to buckling instabilities due to their length compared with their diameter. In some cases rigid walls modify the boundary conditions during the loading or the buckling phenomenon, it will be called conditioned buckling. This paper presents the buckling of a homogeneous cable in a horizontal circular rigid duct subjected to its own weight and an axial compressive load. Buckling load and pitch associated to sinusoidal and helical buckling are determined as transmission of axial load during buckling. A finite element simulation is computed and comparisons are made with the analytical solution. A direct application to the optical fibre cable laying in underground duct is made and experimentation is conducted with a life-sized bench designed at the LM ² S of the ENSAM of Paris and located at the CNET Lannion. Comparisons between experimental and analytical results are presented.     

Elastic lateral-torsional buckling of circular arches subjected to a central concentrated load

An arch under an in-plane central concentrated radial load is subjected to combined axial compressive and bending actions. When these combined axial compressive and bending actions reach a certain value, the arch may suddenly deflect laterally and twist out of its plane of loading and fail in a lateral-torsional buckling mode. This paper derives analytical solutions for the elastic lateral-torsional buckling load of pin-ended circular arches that are subjected to a central concentrated load, using the principle of stationary potential energy in conjunction with the Rayleigh-Ritz method. Analytical solutions of the buckling load for in-plane fixed and out-of-plane pin-ended arches and for the case of the load acting above or below the shear centre are also derived. The analytical solutions are compared with results of a commercial finite element package ANSYS and a finite element code developed by authors elsewhere for arches with different slendernesses, included angles, and cross-sections. The agreement between the analytical solutions and the finite element results is very good.