2 A New Iwan/Palmov Implementation for Fast Simulation and System Identification 19 Table 2.2 Average simulation time for each of the three methods Integration method Simulation time (in seconds) Newmark–beta 214.24 Averaging 1.961 Newmark beta and averaging combined (NB+Averaging) 14.303 0 0.5 1 1.5 2 2.5 Force Amplitude ×105 0 50 100 150 200 250 300 350 simulation time (in s) Newmark-Beta averaging NB+averaging Fig. 2.10 Variation of simulation time with input force amplitude Speed As a measure of speed, the average simulation time was compared for the range of force amplitudes considered. Table 2.2 shows the time taken by each of the three methods being compared to simulate the response. It is evident that the method of averaging speeds up the integration process significantly. This was anticipated as applying the concept of averaging allows for a larger time step to be considered, thus lowering the number of integration steps. For example, for the cases shown in Fig. 2.10, 79,976 time steps were required for the NB method (200 samples per period) whereas for the same duration of interest, with the method of averaging, the RK integrator used an average of 2984 time steps. It can also be observed that varying the amplitude of the impulse force applied does not result in a considerable change in simulation time (2.10). 2.4 Conclusions This work has shown that the method of averaging can be effectively used to speed up the integration of a single degree of freedom system with an Iwan joint with minimal loss in accuracy up to surprisingly large forces. In the micro-slip regime, the damping ratio, natural frequency and response amplitude estimated were found to be comparable to those from the Newmarkbeta method. In fact, in some cases the averaging method was more reliable and accurate (i.e. didn’t require adjusting the ad-hoc convergence tolerance in the Newton loop). It is also preferable if the final goal is to estimate damping and frequency versus time, because no further processing (i.e. with the Hilbert transform) is required to obtain these. However, the averaging algorithm is fundamentally limited to micro-slip only. One can obtain the best of both worlds by using the Newmark-beta method until the external force dies down and the method of averaging can be used thereafter. With this approach the computation time is reduced significantly, but not as much so as when only the averaging method is used (i.e. the computation time is only reduced by a factor of ∼10 rather than a factor of ∼100). Future work will seek to couple this with a suitable optimization tool, allowing one to determine the modal Iwan parameters for a weakly non-linear mode using frequency domain data. Such an approach could be critical in cases in which modal filtering or bandpass filtering is not applicable.
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