12 D. Shetty and M. S. Allen presented by Brake in [10], which uses somewhat similar concepts in an effort to accelerate integration of Iwan joints, while preserving the ability of the joint to capture macro-slip. This work makes use of the analytical formulas for the behavior of an Iwan joint in microslip but limits the solution to the micro-slip regime and uses the averaging method to compute the time response much more quickly. The averaging method is applicable to systems in which the amplitude and phase vary slowly with time [11]. The following integration technique takes advantage of this behavior to improve the simulation time without significantly affecting the accuracy. It also has the advantage that the instantaneous natural frequency and damping are computed in the course of the time integration. The subsequent section explains the algorithm and the relevant theory behind it. It’s applicability for a single degree of freedom system is then examined through a case study and future research is proposed. 2.2 Understanding the Algorithm The algorithm presented in this work considers a single degree of freedom system (shown in Fig. 2.1) consisting of a linear spring, linear damper and a single non-linear Iwan joint. The simulation is divided into two parts: • Integration in the presence of an external impulsive force and • Simulating the free decay response after the external force goes to zero. Matlab’s adaptive fourth–fifth order Runge-Kutta (RK) algorithm (i.e. ode45), is used for both parts. This requires that the differential equation be defined in the form ˙x = f (x, t) with the initial conditions provided as input. It is important to note that the traditional Iwan element cannot be integrated using Runge-Kutta, because it is hysteretic by definition and so it cannot be written in the form described above (without some kind of approximation). For that reason, the Newmarkbeta algorithm has been used with a fixed time step in most cases in the literature. The procedure for obtaining the initial conditions is explained below. 2.2.1 When the External Force is Non-zero The forced equation of motion can be approximated as follows when the Iwan joint remains in the micro-slip regime, ¨x +2ζ(A)ωn(A)˙x +(ωn(A)) 2 x =F(t)/m (2.1) where A is the amplitude of vibration, which is elaborated below. The closed form expressions of dissipation per cycle ( D) and secant stiffness (Kj) derived in [6] are used after converting the physical Iwan parameters [Fs, Kt, χ, β] to their mathematical equivalents [R, S, χ, φmax] D= 4RAχ+3 (χ +3)(χ +2) (2.2) r = A φmax (2.3) Kj =Kt 1− rχ+1 (χ +2)(β +1) (2.4) Fig. 2.1 Schematic of the single degree of freedom system
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