40 M. Wall et al. In analyzing the dynamic response of structures, it is common to represent the friction that occurs at contact surfaces by means of Coulomb friction law. As simple Coulomb models with few parameters oversimplify the description of frictional joints, a continuous model with the ability to include micro-slip seems to be advantageous. Moreover, we should consider that the coefficient of friction is also not constant. It depends primarily on the contact surface properties which change during slip, and on the magnitude of the clamping pressure exerted by the bolts. In [4], it is shown that the coefficient of friction decreases with increasing clamping pressure. According to [5], for “practical surfaces” the coefficient of friction decreases approximately linearly with an increasing normal load. In order to represent micro-slip friction accurately, the finite element model typically requires a mesh of elements having far greater density in the region of the joint compared to the rest of the structure. Computationally, high-fidelity models may be feasible for simple structures that contain one or a few joints, though some simplifications may be employed to speed up the computational time such as model-order reduction of the linear domain away from the joint. Using the best of these approaches, the time required to simulate the response to a dynamic loading is still on the order of hours, and this time would stretch to weeks or months for more realistic structures, such as the turbofan jet engine, which can contain hundreds of bolted joint interfaces. It can be claimed that when a structure vibrates in the shape of one of its modes, the joints dissipate energy and lose stiffness in a manner that is unique to that mode. When the structure is excited to higher response amplitudes, the loss in stiffness and increase in energy dissipation can be observed experimentally as a decrease in the natural frequency and increase in the damping ratio, respectively, for that mode. Both the natural frequency and the damping can be retrieved for each mode using well-established experimental practices and signal processing techniques. The current techniques for extracting the frequency and damping from a model are based on signal processing of the simulated response. Both the signal processing techniques and dynamic simulations are prohibitively expensive to compute, which devalues their use in a practical model updating routine. The problem of computational expense in dynamic simulations was overcome by treating the joint as a quasi-static subcomponent in an otherwise linear, dynamic global model. The groundwork for this solution was laid out in a paper by Festjens et al. [6]. They used the fact that the contact nonlinearity is governed by micro and meso-scale parameters (geometry, roughness, local pressure, etc.) and as a result cannot be included in a macro-size model of a whole structure because of the computational cost. They investigate the idea of using the normal modes of the linearized structure as boundary conditions on a detailed model reduced to the joints only. It has been observed that after a number of repetitive loading cycles, the response of a bolted joint structure may lead to a stabilized state called limit cycle and in the case of an assumed linear structure, this limit cycle is known [6]. Moreover, we can consider the fact that under linear assumption the use of modal coordinates is useful to reduce the size of vibrational problems [7, 8]. After a few modifications to the approach of Festjen et al. [6], Allen et al. [9] later presented a fast and efficient computational method for extracting the amplitude-dependent modal properties from a finite element model and applied it to structures where the joints were modeled as discrete Iwan elements. The new method termed “quasi-static modal analysis” (QSMA), estimates the effective modal natural frequency and damping from a single, static deflection in response to a monotonically-increasing load distributed over the entire structure in the shape of one of its modes. Masing’s rules are then applied to the force-deflection relationship to quantify the amount of energy dissipated and stiffness lost in the all joints when the structure vibrates in the mode of interest [9]. For Allen et al.’s work, Iwan elements were selected since they can capture microslip efficiently, which gives rise to energy dissipation and nonlinear behavior of the joint. However, Iwan elements require four parameter inputs that must be tuned based upon prototype results since they cannot be deduced from first principles but must be measured experimentally [10, 11]. The method of QSMA dramatically reduces the computational effort because the change in the natural frequency and damping of each mode with modal amplitude can be completely determined from a single set of quasi-static, monotonic loading cases. This is in stark contrast to a dynamic simulation, where the free-response history must be computed until its amplitude decays to the smallest level of interest [12]. This advancement prompted Jewell, Allen & Lacayo [13] to apply this technique to detailed finite element models that included nonlinear contact between the bolted interfaces using a commercial software package. While their results showed that such an analysis is feasible, they struggled to obtain accurate results and noted that, once the structure had been meshed with adequate fidelity to capture micro-slip, the computational cost was very significant even to perform a single static analysis. Zare and Allen [14, 15] explored a more computationally efficient alternative that follows the work of Ahn and Barber [16, 17] in that they combined QSMA with a static reduction technique, retaining only those DOF on the interface. They implemented the QSMA for structures where the joint is modeled in detail, using a block Gauss-Seidel algorithm to solve the nonlinear contact problem in Matlab. In this paper, a model of the S4 Beam is constructed in the commercial finite element analysis software package, Abaqus® , with a Coulomb friction law assigned at the interfaces between the parts. The model is loaded using Quasi-Static
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