Optimum transmission of elastic waves through joints

An elastic wave transmitting system (e.g. a nonuniform elastic rod) consisting of an input section with constant impedance, a joint with variable impedance, and an output section with constant impedance is considered. The efficiency of energy transmission (the ratio of transmitted to incident elastic wave energy) is determined and the following optimization problem is solved for two sample cases: Given the properties of the wave transmitting system and a limited duration of the incident wave, determine the shape of this wave such that the efficiency of energy transmission is maximized. In the first case (a joint with constant impedance) the optimization problem leads to a matrix eigenvalue problem. In the second case (a joint with concentrated mass) it leads to an eigenvalue problem for an integral equation. The efficiencies obtained for the optimum incident waves are compared with those obtained for rectagular incident waves an it is found that the differences are generally small. The results, which are discussed with particular reference to the transmission of elastic wave energy through drill rod joints, can also be interpreted for e.g. shallow water waves and electromagnetic waves in a transmission line.     

Shakedown in elastic contact problems with Coulomb friction

Elastic systems with frictional interfaces subjected to periodic loading are sometimes predicted to ‘shake down’ in the sense that frictional slip ceases after the first few loading cycles. The similarities in behaviour between such systems and monolithic bodies with elastic–plastic constitutive behaviour have prompted various authors to speculate that Melan’s theorem might apply to them – i.e., that the existence of a state of residual stress sufficient to prevent further slip is a sufficient condition for the system to shake down.In this paper, we prove this result for ‘complete’ contact problems in the discrete formulation (i) for systems with no coupling between relative tangential displacements at the interface and the corresponding normal contact tractions and (ii) for certain two-dimensional problems in which the friction coefficient at each node is less than a certain critical value. We also present counter-examples for all systems that do not fall into these categories, thus giving a definitive statement of the conditions under which Melan’s theorem can be used to predict whether such a system will shake down.     

Contact with friction of a rigid cylinder with an elastic half-space

An exact solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space is presented. The corresponding boundary-value problem is formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation is constructed using the Wiener-Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full stick between the cylinder and half-space.     

Configuration-dependent friction coefficients and elastic dumbbell rheology

The kinetic theory of nonlinear elastic dumbbells, with bead friction coefficients that depend linearly on the interbead distance, is used to obtain the elongational viscosity and the dumbbell stretching as a function of elongation rate. The results are obtained by solving numerically the “diffusion equation” for the configurational distribution function. No S-shaped curves were found for the elongational viscosity or for the mean-square end-to-end distance. Previous investigators did report S-shaped curves and related “hysteresis” effects. However, their results were based on using mathematical approximations that now appear to be inappropriate.     

The sliding interface crack with friction between elastic and rigid bodies

The paper analyzes the frictional sliding crack at the interface between a semi-infinite elastic body and a rigid one. It gives solutions in complex form for non-homogeneous loading at infinity and explicit solutions for polynomial loading at the interface. It is found that the singularities at the crack tips are different and that they are related to distinct kinematics at the crack tips. Firstly, we postulate that the geometry of the equilibrium crack with crack-tip positions b and a is determined by the conditions of square integrable stresses and continuous displacement at both crack tips. The crack geometry solution is not unique and is defined by any compatible pair (b,a) belonging to a quasi-elliptical curve. Then we prove that, for an equilibrium crack under given applied load, the “energy release rate” G_tip, defined at each crack tip by the J_ε-integral along a semi-circular path, centered at the crack tip, with vanishing radius ε, vanishes. For arbitrarily shaped paths embracing the whole crack, with end points on the unbroken zone, the J-integral is path-independent and has the significance of the rate, with respect to the crack length, of energy dissipated by friction on the crack.     

The boundary element method applied to elastic contact problems with friction

In the current paper the boundary integral equations (BIE) for elastic contact problems with friction are derived from the incremental virtual work principle. After introducing contact conditions of adhesion and slip into BIE all variants on boundary are made to discretize by quadratic isoparametric boundary element. In the current paper not only an auto-increment loading law is presented but also the iterative calculation laws for open, slip and adhesion condition are given. The results of numerical examples are satisfactory.     

Dynamic analysis of the variable stiffness support rotor system with elastic rings

Nonlinear Dynamics - Elastic rings are common rotor supporting structure, which have been widely used in aeroengine rotor support system. However, large inertia force and gyroscopic moment may...     

Unilateral contact between a plate and an elastic foundation

Summary The unilateral frictionless contact between a plate and an elastic subgrade is examined. The foundation is modeled as an elastic half-space or as a Winkler subgrade. The variational formulations more convenient to get approximate solutions are presented. Furthermore the results of a numerical investigation relative to an axisymmetric circular plate are given.

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Sommario Si analizza il problema di contatto unilaterale senza attrito tra una piastra e una fondazione elastica. Per la fondazione si esaminano i modelli di semispazio elastico e di suolo alla Winkler.Vengono presentate le formulazioni variazionali più convenienti ai fini dell'approssimazione delle soluzioni col metodo degli elementi finiti e si illustrano alcuni risultati di un'indagine numerica per il caso di piastra circolare con carico assialsimmetrico.

     

Modeling joints with clearance and friction in multibody dynamic simulation of automotive differentials

Defects in kinematic joints can sometimes highly influence the simulation response of the whole multibody system within which these joints are included. For instance, the clearance, the friction, the lubrication and the flexibility affect the transient behaviour, reduce the component life and produce noise and vibration for classical joints such as prismatics, cylindrics or universal joints. In this work, a new 3D cylindrical joint model which accounts for the clearance, the misalignment and the friction is presented. This formulation has been used to represent the link between the planet gears and the planet carrier in an automotive differential model.     

Indentation of a punch into an elastic strip with friction and adhesion

In the contact interaction between elastic bodies with friction taken into account, the contact region splits, as a rule, into adhesion and sliding regions {xc[1]}. Contact with adhesion and sliding was first considered by L. A. Galin {xc[2]} in the problem of indentation of a punch with a rectilinear foundation into an elastic half-plane, who obtained an approximate solution of this problem [{xc2}, {xc3}]. Galin's problem was further studied in [{xc4}–{xc9}].     

Shakedown limits for an oscillating,elastic rolling contact with Coulomb friction

We examine experimentally and theoretically the effect of frictional shakedown of a three-dimensional elastic rolling contact. Small oscillations of the local normal forces lead to incremental sliding processes within the area of contact. Consequently, this causes a macroscopic slip motion of the two contacting bodies. If the oscillation amplitude is sufficiently small, the frictional slip ceases after the first few loading periods and a safe shakedown occurs. Otherwise the slip motion is continued and the contact fails.     

Effect of friction in wedging of elastic solids

In this paper the contact problem for an elastic wedge of arbitrary angle is considered. It is assumed that the external load is applied to the medium through a rigid wedge and the coefficient of friction between the loading wedge and the elastic solid is constant. The problem is reduced to a singular integral equation of the second kind with the contact pressure as the unknown function. An effective numerical solution of the integral equation is described and the results of three examples are presented. The comparison of these results with those obtained from the frictionless wedge problem indicates that generally friction has the tendency of reducing the peak values of the stress intensity factors calculated at the wedge apex and at the end points of the contact area.This work was supported by NASA-Laugley under the Grant NGR 39-007-011 and by NSF under the Grant GK-42771X.