Sliding friction in the presence of ultrasonic oscillations: superposition of longitudinal oscillations

Summary  It was often observed that friction forces can be reduced significantly if ultrasonic oscillations are superposed to the macroscopic sliding velocity. This phenomenon can be used to improve machining processes by addition of ultrasonic vibration to tools or workpieces, and forms the basis for many processes of ultrasonic machining. On the other hand, ultrasonic vibrations can be used to generate motion. The thrusting force of ultrasonic motors is provided to the rotor through friction. In the present paper, a simple theoretical model for friction in the presence of ultrasonic oscillations is derived theoretically and validated experimentally. The model is capable of predicting the reduction of the macroscopic friction force as a function of the ultrasonic vibration frequency and amplitude and the macroscopic sliding velocity. Received 22 November 2000; accepted for publication 6 February 2001     

3D electroelastic fields in a functionally graded piezoceramic hollow sphere under mechanical and electric loadings

Summary  The problem of a piezoceramic hollow sphere is investigated analytically based on the 3D equations of piezoelasticity. The functionally graded property of the material along the radial direction can be taken arbitrarily in the paper. Displacement and stress functions are introduced, and two independent state equations with variable coefficients are derived. By employing the laminate model, the two state equations are transformed into ones with constant variables from which the state variable solution is easily obtained. Two linear relationships between the state variables at the inner and outer spherical surfaces are established. Numerical calculations are performed for different boundary conditions imposed on the spherical surfaces. Received 28 February 2001; accepted for publication 26 June 2001     

Influence of spring dynamics and friction on a spring-actuated cam system

Summary  The paper investigates two important aspects, friction and spring motion, of the dynamics of a spring-actuated cam system. The characteristics of the friction on the camshaft are analyzed using the nonlinear pendulum experiment, while the parameters of the friction model are estimated using the optimization technique. The analysis reveals that the friction of the camshaft depends on stick–slip, Stribeck effect and viscous damping. Spring elements are found to have much influence on the dynamic characteristics. Hence, they are modeled as four-degrees-of-freedom lumped masses with equivalent springs. The lumped masses and equivalent springs are obtained to match the static stiffness and natural frequency of the actual spring. The appropriateness of the derived friction and spring model are verified by its application to a vacuum circuit-breaker mechanism of the cam-follower type. Received 23 March 2000; accepted for publication 21 November 2000     

Dynamic fracture behavior of a cracked piezoelectric half space under anti-plane mechanical and in-plane electric impact

Summary  The dynamic response of a cracked piezoelectric half-space under anti-plane mechanical and in-plane electric impacting loads is investigated in the present paper. In the study, the crack is assumed parallel to the free surface of the half-space. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in the Laplace transform domain, which are solved numerically. Then, a numerical Laplace inversion is performed and the dynamic stress and electric displacement factors are obtained as functions of time and geometry parameters. The dynamic energy release rate is derived for piezoelectric materials in terms of the electroelastic intensities and is displayed graphically. Received 5 January 2000; accepted for publication 28 June 2000     

Forecasting supercritical and subcritical Hopf bifurcations in aeroelastic systems

A novel method of forecasting bifurcations based on only the observation of the pre-bifurcation regime is proposed. The method is an extension of previous approaches with a focus on oscillatory systems. The method also enables the use of much less measurement data. Numerical results are presented to demonstrate that this new approach predicts the post-bifurcation regime accurately and to explore the robustness of this method to process noise.     

A piezoelectric-elastic body with inhomogeneities and a crack under plane electrical and anti-plane mechanical loads

Summary  In this paper, we study a two-dimensional electroelastic problem of an infinite piezoelectric body with two circular piezoelectric inhomogeneities, one of which contains a crack. We formulate the stress intensity factor (SIF) analytically and investigate it numerically. The problem is solved based on Bueckner's principle, and is reduced to a problem of a singular integral equation of the first kind with respect to the distribution function of screw dislocation. The effect of interaction between the two inhomogeneities and the crack on the electroelastic field as well as the control of the SIF by electrical loads is investigated. Received 18 April 2000; accepted for publication 24 October 2000     

Discontinuity-induced bifurcations in systems with impacts and friction: Discontinuities in the impact law

This paper concerns the non-smooth dynamics of planar mechanical systems with isolated contact in the presence of Coulomb friction. Following Stronge [Impact Mechanics, Cambridge University Press, Cambridge, 2000], a set of closed-form analytic formulae is derived for a rigid-body impact law based on an energetic coefficient of restitution and a resolution of the impact phase into distinct segments of relative slip and stick. Thus, the impact behavior is consistent both with the assumption of Coulomb friction and with the dissipative nature of impacts. The analysis highlights the presence of boundaries between open regions of initial conditions and parameter values corresponding to distinct forms of the impact law and investigates the smoothness properties of the impact law across these boundaries. It is shown how discontinuities in the impact law are associated with discontinuity-induced bifurcations of periodic trajectories, including non-smooth folds and persistence scenarios. Numerical analysis of an example mechanical model is used to illustrate and validate the conclusions.     

Transient response of an insulating crack between dissimilar piezoelectric layers under mechanical and electrical impacts

Summary  The dynamic response of an interface crack between two dissimilar piezoelectric layers subjected to mechanical and electrical impacts is investigated under the boundary condition of electrical insulation on the crack surface by using the integral transform and the Cauchy singular integral equation methods. The dynamic stress intensity factors, the dynamic electrical displacement intensity factor, and the dynamic energy release rate (DERR) are determined. The numerical calculation of the mode-I plane problem indicates that the DERR is more liable to be the token of the crack growth when an electrical load is applied. The dynamic response shows a significant dependence on the loading mode, the material combination parameters as well as the crack configuration. Under a given loading mode and a specified crack configuration, the DERR of an interface crack between piezoelectric media may be decreased or increased by adjusting the material combination parameters. It is also found that the intrinsic mechanical-electrical coupling plays a more significant role in the dynamic fracture response of in-plane problems than that in anti-plane problems. Received 4 September 2001; accepted for publication 23 July 2002 The work was supported by the National Natural Science Foundation under Grant Number 19891180, the Fundamental Research Foundation of Tsinghua University, and the Education Ministry of China.     

Advances in stability,bifurcations and nonlinear vibrations in mechanical systems

Nonlinear Dynamics -     

Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics

Analytical and experimental investigations are performed in order to characterize the dynamic behaviour of a non-smooth rotational oscillator, which exhibits multiple discontinuity boundaries in the phase space. The physical system consists of a rotating body subjected to an elastic restoring force and in contact with one or two rough discs rotating with constant driving velocities. The presence of multiple discontinuity boundaries caused by frictional contacts leads to non-standard bifurcations that are studied by means of a simple mechanical model.A test set-up has then been built to investigate the correctness of modelling of the friction force and the validity of the proposed model for technical applications: the experimental measurements qualitatively and quantitatively capture the basic scenarios anticipated by the model while a strong robustness of the phenomena pointed out by the theoretical analyses has been revealed in the experiments.     

On the modeling and design of piezocomposites with prescribed properties

Summary  Piezoelectric transversely isotropic matrix containing spheroidal piezoelectric inclusions with different properties and of, generally, diverse aspect ratios is considered. A full set of ten effective electrostatic constants is derived, using the method of effective field. The case, when the inclusions are circular cylinders (fibers) is analyzed in detail. The results are compared with those of several earlier works. They constitute the theoretical framework for the design of piezocomposites with prescribed overall properties. Received 3 May 2001; accepted for publication 26 June 2001     

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.     

Modelling of a resonant inertial oscillation within a torus

Summary  We consider the air contained in a pneumatic tyre with the purpose of investigating its inertial oscillations. We model the tyre as a torus limited by a membrane in contact with the ground. According to this model, we prove that the flow within this torus may be considered as one at low Mach number and that it is ruled by oscillations of incompressible rotating fluid. Investigating such inertial oscillations, we show that the geostrophic oscillation is resonant, and we study the resonance phenomenon. Received 6 June 2000; accepted for publication 22 November 2000     

A closure approximation for nematic liquid crystals based on the canonical distribution subspace theory

 A closure approximation for nematic polymers is presented. It approximates the fourth rank order tensor in terms of lower rank tensors, and is derived in the framework of the canonical distribution subspace theory. This approach requires a choice of the class of distributions: Here the set of Bingham distributions is chosen, as already introduced by Chaubal and Leal (1998). The closure is written in a generic frame of reference, and in an explicit form, so that it can be easily implemented. Such formulation also permits studying the closure approximation with continuation tools, which rather completely describe the system dynamics. The predictions can then be compared with those obtained with the exact model. The shear flow is considered as a test, since this flow condition appears to be the most demanding for closure approximations for nematic polymers. Received: 30 November 1999/Accepted: 30 November 1999     

Wrinkling of thick orthotropic sandwich plates under general loading conditions

Summary  This contribution presents an efficient analytical model as well as a FE computation of the critical load, which leads to local stability failure (wrinkling) in sandwich structures. The analytical model assumes an orthotropic face layer and a thick transversely isotropic core. In the last section, a more general core material model is considered. Common core materials (foams and honeycombs) can be described with good accuracy within this model. The main advantage of the solution is the consideration of general loading conditions for the orthotropic face layer as well as in-plane deformations of the core. The results of the FE calculations and the analytical model are in good agreement with each other. Received 7 January 1999; accepted for publication 15 June 1999     

Supercritical flow in manholes with a bend extension

 Bend manholes of sewers with 45° deflection angles are considered. Previous results are reconsidered by introducing the so-called bend extension. It is demonstrated that the discharge capacity of the 45°- and the 90°-bend manholes is identical if the bend extension, a U-shaped straight downstream portion, is added to the 45° manhole. In addition, the effect of the bend cover is analyzed as a general means to suppress shock waves. Both bend extensions and bend covers may be added to existing structures that convey discharge close to or over the design discharge. The hydraulics of bend manholes are analyzed in detail, including the wave profiles along the inner and the outer bend walls, the complete free surface, the recirculation zone, the pressure distributions on the walls and flow in the downstream sewer. Received: 12 March 2001 / Accepted: 10 August 2001     

Effects of dynamic viscoelastic properties on acoustic diffraction by a solid sphere submerged in a viscous fluid

Summary  This study provides a general analysis for scattering of a planar monochromatic compressional sound wave by a homogeneous, isotropic, viscoelastic, solid sphere immersed in an unbounded viscous, heat-nonconducting, compressible fluid. The dynamic viscoelastic properties of the spherical scatterer and the viscosity of the surrounding fluid are rigorously taken into account in the solution of the acoustic-scattering problem. Havriliak–Negami model for viscoelastic material behaviour along with the appropriate wave-harmonic field expansions and the pertinent boundary conditions are employed to develop a closed-form solution in form of infinite series. Subsequently, the associated acoustic quantities such as the scattered far-field pressure directivity pattern, scattered intensity distribution, differential scattering cross section, and the acoustic radiation force are evaluated for given sets of viscoelastic material properties. Numerical results clearly indicate that, in addition to the traditional fluid viscosity-related mechanisms, the dynamic viscoelastic properties of the solid obstacle can be of major significance in sound scattering. Limiting cases are examined and fair agreements with well-known solutions are established. Received 15 January 2002; accepted for publication 2 July 2002 The authors wish to sincerely thank professors Daniel Levesque, Roderic Lakes, Yves Berthelot, S. Temkin, and Andrei Dukhin for valuable and productive consultations on dynamic theory of viscoelasticity and acoustics of (thermo)viscous media.     

Generalized thermoelastic analysis of the stress-focusing effect in a solid cylinder under rotationally asymmetrical instantaneous thermal loading

Summary  This paper investigates the stress-focusing effect in an infinitely long cylinder under rotationally asymmetrical instantaneous thermal loading on the basis of the generalized thermoelastic Lord–Shulman (L-S) and Green–Lindsay (G-L) theories. Combined forms of the governing equations of both theories are given in a cylindrical coordinate system. The two-dimensional generalized thermoelastic problems are solved by numerical inversion of Laplace transform. Calculations have been performed to find distributions of thermal stresses on the basis of the L-S theory. Stress-focusing phenomena under different heating conditions are presented. The effects of thermomechanical coupling and relaxation time on the stress-focusing phenomena as well as the singularity of stresses are discussed. Received 15 November 2000; accepted for publication 15 November 2001     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

Vibration of beams with general boundary conditions due to a moving random load

Summary  The transverse vibrations of elastic homogeneous isotropic beams with general boundary conditions due to a moving random force with constant mean value are analyzed. The boundary conditions considered are: pinned–pinned, fixed–fixed, pinned–fixed, and fixed–free. Based on the Bernoulli beam theory, the problem is described by means of a partial differential equation. Closed-form solutions for the variance and the coefficient of variation of the beam deflection are obtained and compared for three types of force motion: accelerated, decelerated and uniform. The effects of beam damping and speed of the moving force on the dynamic response of beams are studied in detail. Received 3 December 2001; accepted for publication 30 April 2002