Simplified models of bolted joints under harmonic loading

Abstract

Bolted joints are able to provide the majority of damping for an assembled structure when used to connect components. The dynamic frictional contact analysis of a bolted joint under harmonic loading is investigated by the finite element method. Then the detailed finite element model is replaced by a number of Jenkins elements and the Bouc–Wen model by fitting the hysteresis loops produced by the finite element method. These two latter models lead to much reduced computing load. The advantages and drawbacks of these two simplified models are discussed and this understanding is useful in choosing the right simplified model for a bolted joint.

Introduction

In an assembled structure a large proportion of the damping may be provided by the joints. In his review, Beards [1] estimated that up to 90% of the total system damping might be provided by the joints of an assembled structure. The influence of frictional joints can therefore be very significant. However, the nonlinearity of frictional contact at a pair of surfaces in contact makes analysis far from straightforward. Furthermore, the contact condition keeps varying with time under harmonic loading.

The study of friction in structural joints can be traced back to at least Dėn Hartog’s paper on the response of a mass-spring system with a combination of Coulomb friction and viscous damping [2]. Ferri [3] provided an introduction to some of the analytical (notably the harmonic balance method) and numerical methods used to represent systems containing friction. Ferri’s paper, as well as the review by Gaul and Nitsche [4], highlighted a variety of applications of harnessing the dissipative mechanism of frictional joints.

The review of Gaul and Nitsche also provided an introduction to several different models of frictional contact beyond the most basic Coulomb representation. One of the more popular alternative models of dynamic friction reviewed was developed by Canudas de Wit et al. [5]. In this particular model, interconnecting bristles at the interface represent the contact between two surfaces. The six-parameter model proposed is able to model stick–slip vibration, microslip and a variable breakaway force. All these factors can contribute to the response of the joint and the amount of energy dissipated by the structure. Modelling a wide range of response phenomena is necessary in certain control applications. Oden and Martins [6] provided a very thorough presentation of the intricacies of surface representation and its contribution to dynamic friction response.

In the context of structural mechanics however such a detailed knowledge of surface mechanics is not always necessary. Lee et al. [7] used a relatively detailed finite element model of a joint connection to obtain revised natural frequencies and mode shapes for a beam containing a jointed connection. In this case, the only parameter used to describe the contact conditions was the friction coefficient. The influence of the joint was significant with the natural frequencies of the modes being, on average, more than 6% lower than that of the equivalent rigid structure. Modelling a structure using finite elements with several joints all at the same level of detail would be prohibitively costly however. Gaul and Lenz [8] used a reduced parameter Valanis model to represent hysteresis loops generated experimentally. The Valanis model was then used to represent the joints of a complex finite element structure. This allowed the dynamic response of a complicated structure (an antenna) with many bolted joints to be analysed at much reduced cost.

The main aim of this investigation is to produce models that replicated the dissipative behaviour and dynamic response of a bolted joint in isolation. These models are also to provide an insight into the mechanisms that cause frictional damping at an interface. Initially a detailed model of an isolated bolted joint is created using a commercial finite element package. This joint is then subjected to different levels of preloads and applied torques. The time-domain responses are computed and a series of hysteresis loops are obtained in terms of the torque acting at the joint interface and the resulting angular displacements. The torque at the joint interface can be considered hysteretic as it is a function of the instantaneous conditions and historical behaviour of the model [9]. The displacement when the angular velocity is reversed is the historical condition that makes the energy dissipation mechanism “hysteretic”.

The next target is to reproduce the hysteresis loops generated by the finite element model, but within a system of greatly reduced complexity. The time integration of a system containing Jenkins elements is used for this purpose. Comparing the hysteresis loops of both systems, the Jenkins-element model provides very similar dynamic properties of the system and dissipated a similar amount of energy. Therefore, both systems could be considered equivalent in the context required for this investigation. The Bouc–Wen model [9] is also introduced to represent the detailed model of the joint. It is again successful in reproducing the hysteretic behaviour of the finite element model of the joint. The Jenkins-element model and the Bouc–Wen model are compared.

Jenkins elements were used by Gaul and co-workers for representing bolted joints. The Bouc–Wen model, that was initially put forward to represent elasto-plasticity [9] and later used in other circumstances [10], is now introduced in the paper to model bolted joints.