Elsevier

Mechanism and Machine Theory

Volume 37, Issue 9, September 2002, Pages 895-913
Mechanism and Machine Theory

A comparison of revolute joint clearance models in the dynamic analysis of rigid and elastic mechanical systems

https://doi.org/10.1016/S0094-114X(02)00033-2Get rights and content

Abstract

The dynamic response of mechanisms and machines affected by revolute joint clearance is studied in this article. Critical in the precise prediction of the peak values of forces is the contact model being used. A comparison is made between several continuous contact force models and an impact model. The mechanical system is modelled with rigid or elastic bodies. For the impact model a procedure is presented to estimate the maximum contact force during impact. It is shown how the compliance of the links or lubrication of the joint smooths the peak values of the contact forces.

Résumé

La réponse dynamique des mécanismes et des machines en présence de jeu dans les pivots est étudiée dans cet article. La modélisation du contact est un aspect critique pour une prédiction précise des valeurs extrêmes des forces. Plusieurs modèles avec force de contact continue ainsi qu'un modèle d'impact sont comparés. Le système mécanique est modélisé par des corps rigides et flexibles. Pour le modèle d'impact, une procédure d'estimation de la force de contact maximum durant l'impact est présentée. Nous montrons comment la flexibilité des membres du mécanisme et la lubrification des joints réduisent les valeurs extrêmes des forces de contact.

Introduction

Joint clearances due to manufacturing tolerances and wear can seriously affect the dynamic response of mechanical systems. In unlubricated joints it is usually accompanied by rattling, excessive wear and noise, which is caused by peak contact forces. A critical factor in the prediction of the peak forces is the contact model being used. In the past a considerable amount of experimental and theoretical work has been done to study the effect of joint clearances on the dynamic response of mechanical systems. An overview of the English language literature on this subject up to 1980 is given by Haines [1]. In an early German study Hain [2] discusses the effect of radial joint clearance on the forces in an experimental set-up of a scotch-yoke mechanism. After 1980 Soong and Thompson [3] did experimental work on a slider-crank mechanism with revolute joint clearance between the connecting rod and the slider and they made a comparison with calculated results from a rigid-link model. A spatial manipulator with joint clearance was modelled by Kakizaki et al. [4], who included the effects of elastic links and the control system. Deck and Dubowsky [5] published experimental results on a spatial slider-crank mechanism. The possibility of occurrence of chaotic motion was shown by Seneviratne and Earles [6].

This article will focus on the modelling of joint clearances in a computer code environment for dynamic analysis of mechanical systems. First of all the kinematics of a planar journal bearing will be discussed. In the subsequent section two continuous contact force models are treated, a Hertzian contact model with dissipation and a lubricated, hydrodynamic bearing model. Both models are applied in illustrative examples. In Section 4 the basic equations for a discontinuous contact force model (impact with rebound) in a mechanical system are derived. The numerical aspects of such an analysis are treated in depth and an estimate for the maximum contact force during impact is presented. The various contact models are illustrated for a high-speed slider-crank mechanism with a revolute joint clearance between the connecting rod and the slider. The results for the case of rigid links and Herztian contact forces obtained with a finite element based multibody software system are compared with results as presented by Ravn [7].

Section snippets

Joint clearance model

Joint clearance in a planar revolute joint (see Fig. 1) is usually modelled by the introduction of two extra degrees of freedom, the horizontal and the vertical displacements, x and y, of the journal centre relative to the sleeve centre. Since the relative rotation is unconstrained this means that in a non-contact condition no constraints are introduced by the joint. The interaction between the two parts in the joint is solely achieved by normal and tangential contact forces.

The kinematic

Hertzian contact force model with dissipation

Since we are not interested in the shape nor any other detail of the contact region, we need only a global contact model with few parameters. For an unlubricated joint the Hertzian contact force model is an appropriate choice. Whereas the original Hertzian model does not include any energy dissipation, an extension by Lankarani and Nikravesh [8], [9] includes energy loss due to internal damping. In this model the compressive contact force FN in terms of the penetration depth δ=−gN, and velocity

Impulse equations

In the discontinuous contact force model the duration of contact is assumed to be very short in comparison with the time scale of the problem at hand. Under this assumption the change in velocity is instantaneous and we speak of an impact. The velocity jump is enforced by a very high value of the contact force acting only during a small time interval of contact. In the limit case the force is infinite and the time interval is zero. The integral of the force with respect to time over the

Application to a slider-crank mechanism

A slider-crank mechanism is used as an example to illustrate the effect of the different types of joint clearance models. The same mechanism has been used as an example by Ravn [7], which allows us to compare some results. The mechanism, as shown in Fig. 4, consists of a rigid crank of length 0.05 m, a rigid or elastic connecting rod of length 0.12 m and flexural rigidity EI=6.2146×103 Nm2, and a slider. The slider mass and the uniformly distributed mass of the connecting rod are both 0.145 kg.

Conclusion

Three clearance models for a revolute joint have been considered: a continuous contact force model, an impact model and a model for a hydrodynamic lubricated bearing. For the impact model the impulse equations and numerical aspects in handling transitions between different states of motion have been presented; an estimate for the maximal contact force can be made.

The results from the impact model compare well with those from the Hertzian contact force model. Both models can predict the dynamic

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