Simplified models of bolted joints under harmonic loading
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.
Section snippets
Description of finite element model
The finite element model used to simulate the bolted joint and its contact interface was created using ABAQUS finite element software. In so doing no additional in-house programming was required for the investigations carried out. The comprehensive documentation also gave accessibility to the methods used. Notably the ABAQUS suite of programs is used in a wide range of industries and research applications meaning that the methods employed, and results obtained, can be utilised on a far wider
Jenkins elements
To reduce the complexity of the joint and yet maintain its dynamic properties Jenkins elements are used. Each Jenkins element is a spring and a Coulomb element connected in series. The Coulomb element in its sticking state provides a resistive force equal and opposite to the input force to the element. When the friction capacity of the element is passed, i.e., the input force is greater than the coefficient of friction multiplied by the normal force then the element passes into its sliding
Bouc–Wen model
Wen [9] investigated and developed a model to describe a restoring force with hysteresis that was introduced by Bouc [14]. The model can be combined with a nonhysteretic function to describe the restoring force of a system that has yielded to some degree. In bolted joints the analogy with yielding is that of a joint that has started to undergo microslip and complete yield is equivalent to gross slippage in the joint. Wen [9] described the total restoring force in a hysteretic system as
Finite element results and discussion
The finite element work is broken down into two steps. Initially, the effect of varying the torque magnitude applied to the upper bolted member is considered. Secondly, the impact of varying the bolt preload on the dynamic response is considered. The amount, and mechanism, of energy dissipation is investigated by analysing the hysteresis plots of the joint interface and also by studying the time history of the frictional energy losses.
Fig. 3 shows that when considering the hysteresis loops at
Results from the Jenkins-element model
The finite element hysteresis loop chosen for replication by Jenkins elements is created using a 19 kN bolt preload. In these simulations, the 240 N m torque is applied as edge loads to the top block. Before the parameters of the equations describing the joint force could be extracted, the hysteresis loop generated using finite elements has to be discretised into segments of constant stiffness.
The most immediate way to discretise the hysteresis loop is in steps of constant angular displacement or
Results from Bouc–Wen model
It is possible to break down the hysteresis loops into four quadrants each with a different sign combination of velocity and restoring force. In the case of the control hysteresis loop (19 kN preload and 240 N m torque amplitude), the parameter A is obtained from the initial stiffness of the hysteresis response and taken to be 2.1 × 107 N m/rad. To evaluate α and β two points on the hysteresis loop from the finite element method are taken, and from the analytical solutions of z two nonlinear algebraic
Conclusions
Friction behaviour at a bolted interface is a complicated and nonlinear phenomenon. The aim of this investigation was to gain an understanding of the friction phenomena taking place in a bolted joint subjected to dynamic loading. A detailed finite element model of a bolted joint was created to provide the necessary information about the response of the joint to a dynamically applied torque. Then a second-order system was created to represent the dynamic response of the finite element model,
Acknowledgement
The authors gratefully acknowledge the support of the Engineering and Physical Sciences Research Council (EPSRC grant GR/R08520/01).
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